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# User:Gus Wiseman

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*"His subject is the most curious of all— there is none in which truth plays such odd pranks." - G. H. Hardy*

I am formerly a student of pure mathematics at University of California, Davis. This is my primary academic website. To contact me, click "Email this user" to the left (you need to be signed in).

Last updated: Jun 09 2020

## Articles

## Favorite Sequences

- A004111Number of rooted identity trees with n nodes.
- A007097Primeth recurrence: a(n+1) = a(n)-th prime.
- A050376Fermi-Dirac primes.
- A060223Number of normal prime sequences of length n.
- A063834Twice partitioned numbers.
- A076146Matula-Goebel numbers of the finite ordinal numbers.
- A108917Number of knapsack partitions of n.
- A164336All terms are primes raised to the values of earlier terms of the sequence.
- A224751Expansion of Pi in base 27.
- A269134Number of combinatory separations of normal multisets of weight n.
- A273461Number of physically stable n X n placements of water source-blocks in Minecraft.
- A273873Number of strict trees of weight n.
- A275024Total weight of the n-th twice-prime-factored multiset partition.
- A275307Number of labeled spanning blobs on n vertices.
- A276625Finitary numbers. Matula-Goebel numbers of rooted identity trees.
- A277576Rootless recurrence.
- A277996Number of distinct orderless Mathematica expressions with one atom and n positions.
- A279944Number of positions in the free pure symmetric multifunction in one symbol with j-number n.
- A295632Write 1/Product_{n > 1}(1 - 1/n^s) in the form Product_{n > 1}(1 + a (n)/n^s).
- A298426Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

## Open Problems

- A002865Number of partitions of n that do not contain 1 as a part. Conjecture: Also the number of integer partitions of n that have a consecutive subsequence summing to each positive integer from 1 to n.
- A007052Number of order-consecutive partitions of n. Conjecture: Also the number of ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block (A332872).
- A007562Let a(n) be the number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2. Let b(n) be the number of unlabeled rooted trees with n nodes in which every non-leaf node has at least one leaf. Is a(n) = b(n) for all n?
- A028859Number of words of length n without adjacent 0's from the alphabet {0,1,2}. Conjecture: Also the number of length n + 1 sequences covering an initial interval of positive integers whose non-adjacent parts are weakly decreasing.
- A076934Let a(n) be the smallest integer of the form n/k!. Is every positive integer in this sequence?
- A082582Expansion of (1 + x^2 - sqrt( 1 - 4*x + 2*x^2 + x^4)) / (2*x) in powers of x. Conjecture: Also the number of maximal simple graphs with vertices {1...n} and no weakly nesting edges.
- A258409Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n. Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n.
- A261041Number of set partitions of subsets of {1..n} with all consecutive integers in different blocks. Conjecture: Also the number of set partitions of {1..n+1} where, if x and x + 2 belong to the same block, then so does x + 1.
- A275972Conjecture: Let a(n) = A275972(n) be the number of strict knapsack partitions of n. Then a(n) < a(n+1) iff n is even and positive.
- A292444Let a(n) = A292444(n) be the number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets. Is this sequence equal to A258939 or A035948?
- A300574Let a(n) be the coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...). Is this sequence strictly nonnegative? If so, it would be interesting to have a combinatorial interpretation.
- A300789Conjecture: The Young diagram of an integer partition y can be tiled by dominos iff the odd parts of y appear as many times in even as in odd positions. See A000712, A296625, A300060. (Partial credit for this conjecture is due to Wouter Meeussen).
- A304362Let a(n) = Sum_{d|n, d = 1 or not a perfect power} mu(n/d). Up to n = 10^7 this sequence only takes values in {-2, -1, 0, 1, 2}. Is this true in general?
- A307734Let a(n) be the smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k. Conjecture: this sequence has infinitely many non-zero terms.
- A317554Let a(n) be the sum of coefficients in the expansion of p(y) in terms of Schur functions, where p is power-sum symmetric functions and y is the integer partition with Heinz number n. Is this sequence nonnegative? If so, is there a combinatorial interpretation?
- A319055Maximum product of an integer partition of n with relatively prime parts. After a(7), is this the same as A319054?
- A320632Mats Granvik has conjectured that there exists a pair of factorizations of n into factors > 1 where no factor of one divides any factor of the other iff sigma_0(n) - 2 > (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.
- A321982Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number. Conjecture: All terms are nonnegative (verified up to the 5-ladder).
- A321994Number of different chromatic symmetric functions of hypertrees on n vertices. Stanley conjectured that the number of distinct chromatic symmetric functions of trees with n vertices is equal to A000055, i.e., the chromatic symmetric function distinguishes between trees. This has been verified for trees with up to 25 vertices. If it is true in general, does the chromatic symmetric function also distinguish between hypertrees, meaning this sequence would be equal to A035053?
- A325273Let a(n) be the prime omicron of n!. Conjecture: all terms after a(10) = 4 are less than 4.
- A325458Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k. Franklin T. Adams-Watters has conjectured at A049597 that the k-th column gives the coefficients of the sum of Gaussian polynomials [k,m] for m=0...k.
- A326151Numbers whose product of prime indices is twice their sum of prime indices. Conjecture: the only squarefree terms are 65, 154, and 190.
- A326250Number of weakly nesting simple graphs with vertices {1...n}. Conjecture: A006125(n) = a(n) + A000108(n).
- A326254Number of non-capturing set partitions of {1...n}. Conjectured to be equal to A054391, which has g.f. 1 - 2*x^2 / (2*x^2 - 3*x + 1 - sqrt(1 - 2*x - 3*x^2)).
- A327777Prime numbers whose binary indices have integer mean and integer geometric mean. Conjecture: This sequence is infinite.
- A328959a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n). Conjecture: All terms are nonnegative except for a(1) = -1.
- A329395Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths. Conjecture: also numbers k such that the k-th composition in standard order (A066099) is a palindrome.
- A330995Denominator P(n)/Q(n) = A000041(n)/A000009(n). Conjecture: The only 1's occur at n = 0, 1, 2, 7.
- A330996Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n). Conjecture: This sequence is nondecreasing. More generally, the rational sequence A000041(n)/A000009(n) is nondecreasing for n > 5.
- A331022The number of strict integer partitions of n is a power of 2. Conjecture: This sequence is finite. Conjecture: The analogous sequence for non-strict partitions is: 0, 1, 2.
- A331784Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity. Conjecture: A331912(n)/A331784(n) -> 1 as n -> infinity.
- A332278Number of widely totally co-strongly normal integer partitions of n. Is this sequence bounded?
- A335402Numbers n such that the only normal integer partition of n whose run-lengths are a palindrome is (1)^n. Conjecture: The sequence consists of 0, 1, 4, and all primes except 3.

## Original Sequences (3351)

- A335405Number of integer compositions of n with product n.
- A335404Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.
- A335403Totally additive with a(p) = prime(p) for p prime.
- A335402Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.
- A335377Heinz numbers of non-totally co-strong integer partitions.
- A335376Heinz numbers of totally co-strong integer partitions.
- A335375Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.
- A335374Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.
- A335373Numbers k such that the k-th composition in standard order (A066099) is not unimodal.
- A335279Positions of first appearances in A124771 = number of distinct contiguous subsequences of compositions in standard order.
- A335278First index of strictly decreasing prime quartets.
- A335277First index of strictly increasing prime quartets.
- A335241Numbers whose prime indices are not pairwise coprime, where a singleton is not considered coprime unless it is {1}.
- A335240Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).
- A335239Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).
- A335238Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).
- A335237Numbers whose binary indices are not a singleton nor pairwise coprime.
- A335236Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.
- A335235Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.
- A335124Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.
- A335123Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.
- A335122Irregular triangle whose reversed rows are all integer partitions in graded reverse-lexicographic order.
- A334969Heinz numbers of alternately strong integer partitions.
- A334968Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).
- A334967Numbers k such that the every subsequence (not necessarily contiguous) of the k-th composition in standard order (A066099) has a different sum.
- A334966Numbers k such that the k-th composition in standard order (row k of A066099) has weakly decreasing non-adjacent parts.
- A334965Numbers with strictly increasing prime multiplicities.
- A334442Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
- A334441Maximum part of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order; a(0) = 0.
- A334440Number of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.
- A334439Irregular triangle whose rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
- A334438Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
- A334437Heinz number of the n-th reversed integer partition in graded lexicographical order.
- A334436Heinz numbers of all reversed integer partitions sorted first by sum and then reverse-lexicographically.
- A334435Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
- A334434Heinz number of the n-th integer partition in graded lexicographic order.
- A334433Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.
- A334302Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.
- A334301Irregular triangle read by rows where row k is the k-th integer partition, if partitions are sorted first by sum, then by length, and finally lexicographically.
- A334300Number of distinct nonempty subsequences (not necessarily contiguous) in the n-th composition in standard order (A066099).
- A334299Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).
- A334298Numbers whose prime signature is a reversed Lyndon word.
- A334297Length of the Lyndon factorization of the reversed n-th composition in standard order.
- A334274Numbers k such that the k-th composition in standard order is both a necklace and a reversed co-necklace.
- A334273Numbers k such that the k-th composition in standard order is both a reversed necklace and a co-necklace.
- A334272Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.
- A334271Number of compositions of n that are both a reversed necklace and a co-necklace.
- A334270Number of sequences of length n that cover an initial interval of positive integers and are both a reversed Lyndon word and a co-Lyndon word.
- A334269Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.
- A334268Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.
- A334267Numbers k such that the k-th composition in standard order is both a Lyndon word and a reversed co-Lyndon word.
- A334266Numbers k such that the k-th composition in standard order is both a reversed Lyndon word and a co-Lyndon word.
- A334265Numbers k such that the k-th composition in standard order is a reversed Lyndon word.
- A334033The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.
- A334032The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.
- A334031The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.
- A334030Number of combinatory separations of a multiset whose multiplicities are the parts of the n-th composition in standard order.
- A334029Length of the co-Lyndon factorization of the k-th composition in standard order.
- A334028Number of distinct parts in the n-th composition in standard order.
- A333943Numbers k such that the k-th composition in standard order is a reversed necklace.
- A333942Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.
- A333941Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.
- A333940Number of Lyndon factorizations of the k-th composition in standard order.
- A333939Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.
- A333769Irregular triangle read by rows where row k is the sequence of run-lengths of the k-th composition in standard order.
- A333768Minimum part of the n-th composition in standard order. a(0) = 0.
- A333767Length of shortest run of zeros after a one in the binary expansion of n. a(0) = 0.
- A333766Maximum part of the n-th composition in standard order. a(0) = 0.
- A333765Number of co-Lyndon factorizations of the k-th composition in standard order.
- A333764Numbers k such that the k-th composition in standard order is a co-necklace.
- A333755Triangle read by rows where T(n,k) is the number of compositions of n with k runs, n >= 0, 0 <= k <= n.
- A333632Rotational period of the k-th composition in standard order; a(0) = 0.
- A333631Number of permutations of {1..n} with three consecutive terms in arithmetic progression.
- A333630Least STC-number of a composition whose sequence of run-lengths has STC-number n.
- A333629Least k such that the runs-resistance of the k-th composition in standard order is n.
- A333628Runs-resistance of the n-th composition in standard order. Number of steps taking run-lengths to reduce the n-th composition in standard order to a singleton.
- A333627The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.
- A333492Position of first appearance of n in A271410 (LCM of binary indices).
- A333491First index of partially unequal prime quartets.
- A333490First index of unequal prime quartets.
- A333489Numbers k such that the k-th composition in standard order is an anti-run (no adjacent equal parts).
- A333488First index of weakly decreasing prime quartets.
- A333486Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.
- A333485Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically.
- A333484Sort all positive integers, first by sum of prime indices (A056239), then by decreasing number of prime indices (A001222).
- A333483Sort all positive integers, first by sum of prime indices (A056239), then by number of prime indices (A001222).
- A333383First index of weakly increasing prime quartets.
- A333382Number of adjacent unequal parts in the n-th composition in standard-order.
- A333381Number of maximal anti-runs of the n-th composition in standard order.
- A333380Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.
- A333379Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.
- A333257Number of distinct consecutive subsequence-sums of the k-th composition in standard order.
- A333256Numbers k such that the k-th composition in standard order is strictly decreasing.
- A333255Numbers k such that the k-th composition in standard order is strictly increasing.
- A333254Lengths of maximal runs in the sequence of prime gaps (A001223).
- A333253Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).
- A333252Lengths of maximal strictly decreasing subsequences in the sequence of prime gaps (A001223).
- A333231Positions of weak descents in the sequence of differences between primes.
- A333230Positions of weak ascents in the sequence of differences between primes.
- A333229First sums of the Kolakoski sequence A000002.
- A333228Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).
- A333227Numbers k such that the k-th composition in standard order is pairwise coprime, where a singleton is not coprime unless it is (1).
- A333226Least common multiple of the n-th composition in standard order.
- A333225Position of first appearance of n in A333226 stc_lcm (LCMs of compositions in standard order).
- A333224Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.
- A333223Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.
- A333222Numbers k such that every restriction of the k-th composition in standard order to a subinterval has a different sum.
- A333221Irregular triangle read by rows where row n lists the set of STC-numbers of permutations of the prime indices of n.
- A333220The number k such that the k-th composition in standard order consists of the prime indices of n in weakly increasing order.
- A333219Heinz number of the n-th composition in standard order.
- A333218Numbers k such that the k-th composition in standard order is a permutation (of an initial interval).
- A333217Numbers k such that the k-th composition in standard order covers an initial interval of positive integers.
- A333216Lengths of maximal subsequences without adjacent equal terms in the sequence of prime gaps.
- A333215Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).
- A333214Positions of adjacent unequal terms in the sequence of differences between primes.
- A333213Triangle read by rows where T(n,k) is the number of compositions of n with k adjacent terms that are equal or increasing (weak ascents) n >= 0, 0 <= k <= n.
- A333212Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).
- A333195Numbers with three consecutive prime indices in arithmetic progression.
- A333193Number of compositions of n whose non-adjacent parts are strictly decreasing.
- A333192Number of compositions of n with strictly increasing run-lengths.
- A333191Number of compositions of n whose run-lengths are either strictly increasing or strictly decreasing.
- A333190Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.
- A333150Number of strict compositions of n whose non-adjacent parts are strictly decreasing.
- A333149Number of strict compositions of n that are neither increasing nor decreasing.
- A333148Number of compositions of n whose non-adjacent parts are weakly decreasing.
- A333147Number of compositions of n that are either strictly increasing or strictly decreasing.
- A333146Number of non-unimodal negated permutations of the multiset of prime indices of n.
- A333145Number of unimodal negated permutations of the multiset of prime indices of n.
- A332875Sizes of maximal weakly increasing subsequences of A000002.
- A332874Number of strict compositions of n that are neither unimodal nor is their negation.
- A332873Number of non-unimodal, non-co-unimodal sequences of length n covering an initial interval of positive integers.
- A332872Number of ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.
- A332870Number of compositions of n that are neither unimodal nor is their negation.
- A332836Number of compositions of n whose run-lengths are weakly increasing.
- A332835Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.
- A332834Number of compositions of n that are neither weakly increasing nor weakly decreasing.
- A332833Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
- A332832Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.
- A332831Numbers whose unsorted prime signature is neither weakly increasing nor weakly decreasing.
- A332746Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.
- A332745Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.
- A332744Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.
- A332743Number of non-unimodal compositions of n covering an initial interval of positive integers.
- A332742Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.
- A332741Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.
- A332728Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.
- A332727Number of compositions of n whose run-lengths are not unimodal.
- A332726Number of compositions of n whose run-lengths are unimodal.
- A332725Heinz numbers of integer partitions whose negated first differences are not unimodal.
- A332724Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.
- A332673Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.
- A332672Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.
- A332671Number of non-unimodal permutations of the multiset of prime indices of n.
- A332670Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.
- A332669Number of compositions of n whose negation is not unimodal.
- A332668Number of strict integer partitions of n without three consecutive parts in arithmetic progression.
- A332643Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.
- A332642Numbers whose negated unsorted prime signature is not unimodal.
- A332641Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
- A332640Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.
- A332639Number of integer partitions of n whose negated run-lengths are not unimodal.
- A332638Number of integer partitions of n whose negated run-lengths are unimodal.
- A332579Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.
- A332578Number of compositions of n whose negation is unimodal.
- A332577Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.
- A332576Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.
- A332340Number of widely alternately co-strongly normal compositions of n.
- A332339Number of alternately co-strong reversed integer partitions of n.
- A332338Number of alternately co-strong compositions of n.
- A332337Number of widely totally strongly normal compositions of n.
- A332336Number of narrowly totally strongly normal compositions of n.
- A332297Number of narrowly totally strongly normal integer partitions of n.
- A332296Number of narrowly totally normal compositions of n.
- A332295Number of widely recursively normal integer partitions of n.
- A332294Number of unimodal permutations of a multiset whose multiplicities are the prime indices of n.
- A332293Heinz numbers of widely totally co-strongly normal integer partitions.
- A332292Number of widely alternately strongly normal integer partitions of n.
- A332291Heinz numbers of widely totally strongly normal integer partitions.
- A332290Heinz numbers of widely alternately co-strongly normal integer partitions.
- A332289Number of widely alternately co-strongly normal integer partitions of n.
- A332288Number of unimodal permutations of the multiset of prime indices of n.
- A332287Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.
- A332286Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
- A332285Number of strict integer partitions of n whose first differences (assuming the last part is zero) are unimodal.
- A332284Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
- A332283Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.
- A332282Numbers whose unsorted prime signature is not unimodal.
- A332281Number of integer partitions of n whose run-lengths are not unimodal.
- A332280Number of integer partitions of n with unimodal run-lengths.
- A332279Number of widely totally normal compositions of n.
- A332278Number of widely totally co-strongly normal integer partitions of n.
- A332277Number of widely totally normal integer partitions of n.
- A332276Heinz numbers of widely totally normal integer partitions.
- A332275Number of totally co-strong integer partitions of n.
- A332274Number of totally strong compositions of n.
- A332273Sizes of maximal weakly decreasing subsequences of A000002.
- A332272Number of narrowly recursively normal integer partitions of n.
- A331995Numbers with at most one distinct prime prime index.
- A331994Matula-Goebel numbers of semi-lone-child-avoiding rooted semi-identity trees.
- A331993Number of semi-lone-child-avoiding rooted semi-identity trees with n vertices.
- A331992Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.
- A331991Number of semi-lone-child-avoiding achiral rooted trees with n vertices.
- A331967Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.
- A331966Number of lone-child-avoiding rooted semi-identity trees with n unlabeled vertices.
- A331965Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.
- A331964Number of semi-lone-child-avoiding rooted identity trees with n vertices.
- A331963Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.
- A331937a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).
- A331936Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).
- A331935Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.
- A331934Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.
- A331933Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.
- A331916Numbers with exactly one distinct prime prime index.
- A331915Numbers with exactly one prime prime index, counted with multiplicity.
- A331914Numbers with at most one prime prime index, counted with multiplicity.
- A331913Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.
- A331912Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.
- A331875Number of enriched identity p-trees of weight n.
- A331874Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.
- A331873Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.
- A331872Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.
- A331871Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.
- A331785Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.
- A331784Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.
- A331783Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.
- A331687Number of locally disjoint enriched p-trees of weight n.
- A331686Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
- A331685Number of tree-factorizations of Heinz numbers of integer partitions of n.
- A331684Number of locally disjoint enriched identity p-trees of weight n.
- A331683One and all numbers of the the form 2^k * prime(p) for k > 0 and p already in the sequence.
- A331682One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.
- A331681One, two, and all numbers of the the form 2^k * prime(p) where k > 0 and p already belongs to the sequence.
- A331680Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.
- A331679Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.
- A331678Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.
- A331581Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.
- A331580Smallest number whose unsorted prime signature is the reversed unsorted prime signature of n.
- A331579Position of first appearance of n in A124758 stc_prod (products of compositions in standard order).
- A331578Number of labeled series-reduced rooted trees with n vertices and more than two branches of the root.
- A331577Number of labeled rooted trees with n vertices and more than two branches of the root.
- A331490Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).
- A331489Matula-Goebel numbers of topologically series-reduced rooted trees.
- A331488Number of unlabeled series-reduced rooted trees with n vertices and more than two branches (of the root).
- A331418If A331417(n) is the maximum sum of primes of the parts of an integer partition of n, then a(n) = A331417(n) - n + 1.
- A331417Maximum sum of primes of the parts of an integer partition of n.
- A331416Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = k.
- A331387Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.
- A331386Numbers with at least one prime prime index.
- A331385Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = n + k.
- A331384Numbers whose sum of prime factors is equal to their product of prime indices.
- A331383Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.
- A331382Numbers whose sum of prime factors is divisible by their product of prime indices.
- A331381Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.
- A331380Numbers whose sum of prime factors is divisible by their sum of prime indices.
- A331379Number of integer partitions of n whose sum of primes of parts is divisible by n.
- A331378Numbers whose product of prime indices is divisible by their sum of prime factors.
- A331233Number of unlabeled rooted trees with n vertices and more than two branches of the root.
- A331232Record numbers of factorizations into distinct factors > 1.
- A331231Numbers k such that the number of factorizations of k into distinct factors > 1 is even.
- A331230Numbers k such that the number of factorizations of k into distinct factors > 1 is odd.
- A331201Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.
- A331200Least number with each record number of factorizations into distinct factors > 1.
- A331198Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).
- A331051Numbers whose number of factorizations into factors > 1 (A001055) is even.
- A331050Positive integers whose number of factorizations into factors > 1 (A001055) is odd.
- A331049Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.
- A331048Nearest integer to A001055(n)/A045778(n), where A001055 is factorizations and A045778 is strict factorizations.
- A331024Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).
- A331023Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).
- A331022Numbers k such that the number of strict integer partitions of k is a power of 2.
- A330998Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.
- A330997Sorted list containing the least number with each possible nonzero number of factorizations into distinct factors > 1.
- A330996Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).
- A330995Denominator P(n)/Q(n) = A000041(n)/A000009(n).
- A330994Numerator of P(n)/Q(n) = A000041(n)/A000009(n).
- A330993Numbers n such that a multiset whose multiplicities are the prime indices of n has a prime number of multiset partitions.
- A330992Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.
- A330991Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).
- A330990Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).
- A330989Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.
- A330977Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.
- A330976Numbers that are not the number of factorizations into factors > 1 of any positive integer.
- A330975Numbers that are not the number of factorizations of n into distinct factors > 1 for any n.
- A330974Least positive integer with n factorizations into distinct factors > 1, and 0 if no such number exists.
- A330973Least positive integer with exactly n factorizations into factors > 1, and 0 if no such number exists.
- A330972Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.
- A330954Number of integer partitions of n whose product is divisible by the sum of primes of their parts.
- A330953Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.
- A330952Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.
- A330951Number of singleton-reduced unlabeled rooted trees with n nodes.
- A330950Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.
- A330949Odd nonprime numbers whose prime indices are not all prime numbers.
- A330948Nonprime numbers whose prime indices are not all prime numbers.
- A330947Nonprime numbers whose prime indices are all prime numbers.
- A330946Odd numbers whose prime indices are not all prime numbers.
- A330945Numbers whose prime indices are not all prime numbers.
- A330944Number of nonprime prime indices of n.
- A330943Matula-Goebel numbers of singleton-reduced rooted trees.
- A330937Number of strictly recursively normal integer partitions of n.
- A330936Number of nontrivial factorizations of n into factors > 1.
- A330935Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.
- A330785Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.
- A330784Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.
- A330783Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
- A330728Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.
- A330727Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.
- A330726Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.
- A330679Number of balanced reduced multisystems whose atoms constitute an integer partition of n.
- A330677Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.
- A330676Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.
- A330675Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.
- A330668Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.
- A330667Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose atoms are the prime indices of n.
- A330666Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.
- A330665Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.
- A330664Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.
- A330663Number of non-isomorphic balanced reduced multisystems of weight n and depth n (the maximum).
- A330655Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.
- A330654Number of series/singleton-reduced rooted trees on normal multisets of size n.
- A330628Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint).
- A330627Number of non-isomorphic phylogenetic trees with n nodes.
- A330626Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.
- A330625Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.
- A330624Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.
- A330475Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.
- A330474Number of non-isomorphic balanced reduced multisystems of weight n.
- A330473Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.
- A330472Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).
- A330471Number of series/singleton-reduced rooted trees on strongly normal multisets of size n.
- A330470Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.
- A330469Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.
- A330467Number of series-reduced rooted trees whose leaves are multisets whose multiset union is a strongly normal multiset of size n.
- A330465Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.
- A330464Number of non-isomorphic weight-n sets of set-systems with distinct multiset unions.
- A330463Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.
- A330462Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.
- A330461Array read by antidiagonals (upwards) where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.
- A330460Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.
- A330459QQQQ. Number of set partitions of set-systems with total sum n.
- A330458PPQP. Number of multisets of nonempty sets of nonempty multisets of positive integers with total sum n.
- A330457PPPQ. Number of multisets of nonempty multisets of nonempty sets of positive integers with total sum n.
- A330456PPQQ. Number of multisets of nonempty sets of nonempty sets of positive integers with total sum n.
- A330455PQPQ. Number of sets of nonempty multisets of nonempty sets of positive integers with total sum n.
- A330454PQQP. Number of sets of nonempty sets of nonempty multisets of positive integers with total sum n.
- A330453PQPP. Number of strict multiset partitions of multiset partitions of integer partitions of n.
- A330452QPPP. Number of set partitions of strict multiset partitions of integer partitions of n.
- A330417Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.
- A330415Coefficient of h(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where h is the basis of homogeneous symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.
- A330346Number of unlabeled simple graphs covering n vertices with exactly two automorphisms.
- A330345Number of labeled simple graphs with n vertices whose covered portion has exactly two automorphisms.
- A330344Number of unlabeled graphs with n vertices whose covered portion has exactly two automorphisms.
- A330343Number of labeled fully chiral simple graphs (also called identity or asymmetric graphs) covering n vertices.
- A330297Number of labeled simple graphs covering n vertices with exactly two automorphisms, or with exactly n!/2 graphs obtainable by permuting the vertices.
- A330296BII-numbers of set partitions with at least two blocks.
- A330295Number of non-isomorphic fully chiral set-systems covering n vertices.
- A330294Number of non-isomorphic fully chiral set-systems on n vertices.
- A330282Number of fully chiral set-systems on n vertices.
- A330281Numbers whose prime-indices do not have weakly increasing numbers of distinct prime factors.
- A330236MM-numbers of fully chiral multisets of multisets.
- A330235Number of fully chiral factorizations of n.
- A330234Number of achiral factorizations of n into factors > 1.
- A330233Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).
- A330232MM-numbers of achiral multisets of multisets.
- A330231Number of distinct set-systems that can be obtained by permuting the vertices of the set-system with BII-number n.
- A330230Least MM-number of a multiset of multisets with n distinct representatives obtainable by permuting the vertices.
- A330229Number of fully chiral set-systems covering n vertices.
- A330228Number of fully chiral integer partitions of n.
- A330227Number of non-isomorphic fully chiral multiset partitions of weight n.
- A330226BII-numbers of fully chiral set-systems.
- A330225Position of first appearance of n in A290103 = LCM of prime indices.
- A330224Number of achiral integer partitions of n.
- A330223Number of non-isomorphic achiral multiset partitions of weight n.
- A330218Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.
- A330217BII-numbers of achiral set-systems.
- A330216Number of strict integer partitions of n whose product is a powerful number.
- A330196Number of unlabeled set-systems covering n vertices with no endpoints.
- A330195BII-number of the BII-normalization of the set-system with BII-number n.
- A330194MM-number of the MM-normalization of the multiset of multisets with MM-number n.
- A330124Number of unlabeled set-systems with n vertices and no endpoints.
- A330123BII-numbers of MM-normalized set-systems.
- A330122MM-numbers of MM-normalized multiset partitions.
- A330121MM-numbers of lexicographically normalized multiset partitions.
- A330120MM-numbers of lexicographically normalized multisets of multisets.
- A330110BII-numbers of lexicographically normalized set-systems.
- A330109BII-numbers of BII-normalized set-systems.
- A330108MM-numbers of MM-normalized multisets of multisets.
- A330107MM-numbers of brute-force normalized multiset partitions.
- A330106Number of integer partitions of n whose product is a powerful number.
- A330105MM-number of the brute-force normalization of the multiset of multisets with MM-number n.
- A330104MM-numbers of brute-force normalized multisets of multisets.
- A330103Numbers whose prime-indices do not have weakly increasing numbers of prime factors, counted with multiplicity.
- A330102BII-number of the VDD-normalization of the set-system with BII-number n.
- A330101BII-number of the brute-force normalization of the set-system with BII-number n.
- A330100BII-numbers of VDD-normalized set-systems.
- A330099BII-numbers of brute-force normalized set-systems.
- A330098Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.
- A330097MM-numbers of VDD-normalized multiset partitions.
- A330061MM-number of the VDD-normalization of the multiset of multisets with MM-number n.
- A330060MM-numbers of VDD-normalized multisets of multisets.
- A330059Number of set-systems with n vertices and no endpoints.
- A330058Number of non-isomorphic multiset partitions of weight n with at least one endpoint.
- A330057Number of set-systems covering n vertices with no singletons or endpoints.
- A330056Number of set-systems with n vertices and no singletons or endpoints.
- A330055Number of non-isomorphic set-systems of weight n with no singletons or endpoints.
- A330054Number of non-isomorphic set-systems of weight n with no endpoints.
- A330053Number of non-isomorphic set-systems of weight n with at least one singleton.
- A330052Number of non-isomorphic set-systems of weight n with at least one endpoint.
- A330029Numbers whose binary expansion has cuts-resistance <= 2.
- A330028Number of compositions of n with cuts-resistance <= 2.
- A329871Number of static n X n placements of water source-blocks in Minecraft.
- A329870Runs-resistance of the binary expansion of n without the first digit.
- A329869Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.
- A329868Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.
- A329867Runs-resistance minus cuts-resistance of the binary expansion of n.
- A329866Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.
- A329865Numbers whose binary expansion has the same runs-resistance as cuts-resistance.
- A329864Number of compositions of n with the same runs-resistance as cuts-resistance.
- A329863Number of compositions of n with cuts-resistance 2.
- A329862Positive integers whose binary expansion has cuts-resistance 2.
- A329861Triangle read by rows where T(n,k) is the number of compositions of n with cuts-resistance k.
- A329860Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.
- A329768Number of finite sequences of positive integers whose sum minus runs-resistance is n.
- A329767Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.
- A329766Number of compositions of n whose run-lengths cover an initial interval of positive integers.
- A329750Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.
- A329749Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.
- A329748Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.
- A329747Runs-resistance of the sequence of prime indices of n.
- A329746Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.
- A329745Number of compositions of n with runs-resistance 2.
- A329744Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.
- A329743Number of compositions of n with runs-resistance n - 3.
- A329741Number of compositions of n whose multiplicities cover an initial interval of positive integers.
- A329740Number of compositions of n whose multiplicities are distinct and cover an initial interval of positive integers.
- A329739Number of compositions of n whose run-lengths are all different.
- A329738Number of compositions of n whose run-lengths are all equal.
- A329661BII-number of the set-system whose MM-number is A329629(n).
- A329632Number of connected integer partitions of n whose distinct parts are pairwise indivisible.
- A329631Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.
- A329630Products of distinct primes of nonprime squarefree index.
- A329629Products of distinct odd primes of squarefree index.
- A329628Smallest BII-number of an intersecting antichain with n edges.
- A329627Smallest BII-number of a clutter (connected antichain) with n edges.
- A329626Smallest BII-number of an antichain with n edges.
- A329625Smallest BII-number of a connected set-system with n edges.
- A329561BII-numbers of intersecting antichains of sets.
- A329560BII-numbers of antichains of sets with empty intersection.
- A329559MM-numbers of multiset clutters (connected weak antichains of multisets).
- A329558Product of primes indexed by the first n squarefree numbers.
- A329557Smallest MM-number of a set of n nonempty sets.
- A329556Smallest MM-number of a set of n sets with no singletons.
- A329555Smallest MM-number of a clutter (connected antichain) of n distinct sets.
- A329554Smallest MM-number of a set of n nonempty sets with no singletons.
- A329553Smallest MM-number of a connected set of n multisets.
- A329552Smallest MM-number of a connected set of n sets.
- A329401Numbers whose binary expansion without the most significant (first) digit is a co-Lyndon word.
- A329400Length of the co-Lyndon factorization of the binary expansion of n with the most significant (first) digit removed.
- A329399Numbers whose reversed binary expansion has uniform Lyndon factorization.
- A329398Number of compositions of n with uniform Lyndon factorization and uniform co-Lyndon factorization.
- A329397Number of compositions of n whose Lyndon factorization is uniform.
- A329396Numbers n such that the co-Lyndon factorization of the binary expansion of n is uniform.
- A329395Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.
- A329394Number of compositions of n whose Lyndon and co-Lyndon factorizations both have the same length.
- A329367Numbers whose binary expansion, without the most significant digit, is not a necklace.
- A329366Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).
- A329362Length of the co-Lyndon factorization of the first n terms of A000002.
- A329361a(n) = Sum_{i = 1..n} 2^(n - i) * A000002(i).
- A329360The decimal expansion of a(n) is the first n terms of A000002.
- A329359Irregular triangle read by rows where row n gives the lengths of the factors in the co-Lyndon factorization of the binary expansion of n.
- A329358Numbers whose binary expansion has Lyndon and co-Lyndon factorizations of equal lengths.
- A329357Numbers whose reversed binary expansion has co-Lyndon factorization of length 2.
- A329356The binary expansion of a(n) is the first n terms of 2 - A000002.
- A329355The binary expansion of a(n) is the second through n-th terms of A000002 - 1.
- A329327Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).
- A329326Length of the co-Lyndon factorization of the reversed binary expansion of n.
- A329325Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.
- A329324Number of Lyndon compositions of n whose reverse is not a co-Lyndon composition.
- A329318List of co-Lyndon words on {1,2} sorted first by length and then lexicographically.
- A329317Length of the Lyndon factorization of the reversed first n terms of A000002.
- A329316Irregular triangle read by rows where row n gives the sequence of lengths of components of the Lyndon factorization of the reversed first n terms of A000002.
- A329315Irregular triangle read by rows where row n gives the sequence of lengths of components of the Lyndon factorization of the first n terms of A000002.
- A329314Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n.
- A329313Length of the Lyndon factorization of the reversed binary expansion of n.
- A329312Length of the co-Lyndon factorization of the binary expansion of n.
- A329145Number of non-necklace compositions of n.
- A329144Number of integer partitions of n whose differences are a periodic word.
- A329143Number of integer partitions of n whose augmented differences are a periodic word.
- A329142Numbers whose prime signature is not a necklace.
- A329141Number of Lyndon compositions of n that are not weakly increasing.
- A329140Numbers whose prime signature is a periodic word.
- A329139Numbers whose prime signature is an aperiodic word.
- A329138Numbers whose prime signature is a necklace.
- A329137Number of integer partitions of n whose differences are an aperiodic word.
- A329136Number of integer partitions of n whose augmented differences are an aperiodic word.
- A329135Numbers whose differences of prime indices are an aperiodic word.
- A329134Numbers whose differences of prime indices are a periodic word.
- A329133Numbers whose augmented differences of prime indices are an aperiodic sequence.
- A329132Numbers whose augmented differences of prime indices are a periodic sequence.
- A329131Numbers whose prime signature is a Lyndon word.
- A328966Number of strict factorizations of n with integer average.
- A328965Smallest k such that (omega(k) - 1) * nu(k) = n, and 0 if none exists, where nu = A001221, omega = A001222.
- A328964Smallest k such that omega(k) * nu(k) = n, and 0 if none exists, where nu = A001221, omega = A001222.
- A328963Smallest k such that sigma_0(k) - 2 - (omega(k) - 1) * nu(k) = n, and 0 if none exists, where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328962Smallest k such that sigma_0(k) - omega(k) * nu(k) = n, and 0 if none exists, where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328961Positive integers n such that sigma_0(n) - 3 = (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328960Number of integer partitions of n whose number of nontrivial submultisets is greater than their number of distinct parts times their number of parts minus 1.
- A328959a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328958a(n) = sigma_0(n) - omega(n) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328957Numbers n such that sigma_0(n) != omega(n) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328956Numbers n such that sigma_0(n) = omega(n) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.
- A328871Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).
- A328870Numbers whose lengths of runs of 1's in their reversed binary expansion are not weakly increasing.
- A328869Numbers whose lengths of runs of 1's in their reversed binary expansion are weakly increasing.
- A328868Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.
- A328867Heinz numbers of integer partitions in which no two distinct parts are relatively prime.
- A328679Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.
- A328678Number of strict, pairwise indivisible, relatively prime integer partitions of n.
- A328677Numbers whose distinct prime indices are relatively prime and pairwise indivisible.
- A328676Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible.
- A328675Number of integer partitions of n with no two distinct consecutive parts divisible.
- A328674Numbers whose distinct prime indices have no consecutive divisible parts.
- A328673Number of integer partitions of n in which no two distinct parts are relatively prime.
- A328672Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.
- A328671Numbers whose binary indices are relatively prime and pairwise indivisible.
- A328670Number of aperiodic compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.
- A328669Number of Lyndon compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.
- A328668Numbers whose binary expansion, without the most significant digit, is a necklace.
- A328609Number of compositions of n whose circularly adjacent parts are relatively prime.
- A328608Numbers whose binary indices have no part circularly followed by a divisor or a multiple.
- A328607Numbers whose reversed binary expansion, without the most significant digit, is a necklace.
- A328603Numbers whose prime indices have no consecutive divisible parts, meaning no prime index is a divisor of the next-smallest prime index, counted with multiplicity.
- A328602Number of necklace compositions of n where no pair of circularly adjacent parts is relatively prime.
- A328601Number of necklace compositions of n with no part circularly followed by a divisor or a multiple.
- A328600Number of necklace compositions of n with no part circularly followed by a divisor.
- A328599Number of compositions of n with no part circularly followed by a divisor or a multiple.
- A328598Number of compositions of n with no part circularly followed by a divisor.
- A328597Number of necklace compositions of n where every pair of circularly adjacent parts is relatively prime.
- A328596Numbers whose reversed binary expansion is a Lyndon word (aperiodic necklace).
- A328595Numbers whose reversed binary expansion is a necklace.
- A328594Numbers whose binary expansion is aperiodic.
- A328593Numbers whose binary indices have no consecutive divisible parts.
- A328592Numbers whose binary expansion has all different lengths of runs of 1's.
- A328514MM-numbers of connected sets of sets.
- A328513Connected squarefree numbers.
- A328512Number of distinct connected components of the multiset of multisets with MM-number n.
- A328511Number of non-singleton runs of divisors of 2n.
- A328510Smallest number whose divisors have n non-singleton runs.
- A328509Number of non-unimodal sequences of length n covering an initial interval of positive integers.
- A328508Number of compositions of n with no part divisible by the next or the prior.
- A328460Number of compositions of n with no part divisible by the next.
- A328459Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.
- A328458Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.
- A328457Length of the longest run of divisors > 1 of n.
- A328456LCM of the prime indices of 2n + 1, all minus 1; a(0) = 0.
- A328451Sorted positions of first appearances in A328219, where if n = A000040(i_1) * ... * A000040(i_k), then A328219(n) = LCM(1+i_1,...,1+i_k).
- A328450Numbers that are a smallest number with k pairs of successive divisors, for some k.
- A328449Smallest number in whose divisors the longest run is of length n, and 0 if none exists.
- A328448Smallest number whose divisors > 1 have a longest run of length n, and 0 if none exists.
- A328338Numbers whose third-largest divisor is prime.
- A328337The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).
- A328336Numbers with no consecutive prime indices relatively prime.
- A328335Numbers whose consecutive prime indices are relatively prime.
- A328221Number of integer partitions of n with at least one pair of consecutive divisible parts.
- A328220Number of strict integer partitions of n with no pair of consecutive parts relatively prime.
- A328219LCM of the prime indices of n, all plus 1.
- A328195Maximum length of a divisibility chain of consecutive divisors of n greater than 1.
- A328194Maximum length of a divisibility chain of consecutive nontrivial divisors of n (greater than 1 and less than n).
- A328193Irregular triangle read by rows where row n lists the nontrivial divisors of n (greater than 1 and less than n).
- A328189Numbers n with at least one pair of consecutive divisible nontrivial divisors (greater than 1 and less than n).
- A328188Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.
- A328187Number of integer partitions of n with no pair of consecutive parts relatively prime.
- A328172Number of integer partitions of n with all pairs of consecutive parts relatively prime.
- A328171Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.
- A328170Number of integer partitions of n whose parts minus 1 are relatively prime.
- A328169GCD of the prime indices of n, all plus 1.
- A328168Numbers whose prime indices minus 1 are relatively prime.
- A328167GCD of the prime indices of n, all minus 1.
- A328166Heinz number of the run-lengths of the divisors of n.
- A328165Numbers whose divisors do not have weakly decreasing run-lengths.
- A328164Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.
- A328163Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.
- A328162Maximum length of a divisibility chain of consecutive divisors of n.
- A328161Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.
- A328028Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.
- A328027Irregular triangle read by rows where row n lists, in weakly increasing order, the differences between consecutive divisors of n.
- A328026Number of divisible pairs of consecutive divisors of n.
- A328025Irregular triangle read by rows where row n gives the differences between consecutive divisors of n in weakly decreasing order.
- A328024Heinz numbers of multisets representing the differences between some positive integer's consecutive divisors.
- A328023Heinz number of the multiset of differences between consecutive divisors of n.
- A327908Nonprime squarefree numbers whose prime indices all have the same Omega (A001222) and the same sum of prime indices (A056239).
- A327907Numbers with more than one factorization into at factors > 1 with integer mean.
- A327906Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).
- A327905FDH numbers of pairwise coprime sets.
- A327904Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.
- A327903Number of set-systems covering n vertices where every edge has a different sum.
- A327902Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).
- A327901Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).
- A327900Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).
- A327899Number of set partitions of {1..n} with equal block sizes and equal block sums.
- A327808Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.
- A327807Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.
- A327806Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.
- A327805Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.
- A327784Heinz numbers of integer partitions whose LCM is greater than their sum.
- A327783Heinz numbers of integer partitions whose LCM is a multiple of their sum.
- A327782Numbers n that cannot be written as a sum of two or more distinct parts with the largest part dividing n.
- A327781Number of integer partitions of n whose LCM is less than n.
- A327780Number of integer partitions of n whose LCM is 2 * n.
- A327779Number of integer partitions of n whose LCM is greater than n.
- A327778Number of integer partitions of n whose LCM is a multiple of n.
- A327777Prime numbers whose binary indices have integer mean and integer geometric mean.
- A327776Heinz numbers of integer partitions whose LCM is less than their sum.
- A327775Heinz numbers of integer partitions whose LCM is twice their sum.
- A327695Number of non-constant factorizations of n whose distinct factors are pairwise coprime.
- A327685Nonprime numbers whose prime indices have a common divisor > 1.
- A327659Number of factorizations of A318978(n - 1), the n-th number that is 1 or whose prime indices have a common divisor > 1, into numbers > 1 satisfying the same conditions.
- A327658Number of factorizations of n that are empty or whose factors have a common divisor > 1.
- A327657Number of divisors of n that are 1 or whose prime indices have a common divisor > 1.
- A327656Maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.
- A327540Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.
- A327538Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, prime, or whose prime indices are relatively prime (A327535, A327537).
- A327537Quotient of n over the maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.
- A327536Number of divisors of n that are 1, prime, or whose prime indices are relatively prime.
- A327535Maximum divisor of n that is 1, prime, or whose prime indices are relatively prime.
- A327534Numbers that are 1, prime, or whose prime indices are relatively prime.
- A327533Number of factorizations of the n-th number that is 1 or whose prime indices are relatively prime A289509(n - 1) into numbers > 1 satisfying the same conditions.
- A327532Indicator function for numbers whose prime indices are relatively prime (A289509).
- A327531a(n) = 1 if the prime indices of n are relatively prime, otherwise a(n) = n.
- A327530Number of divisors of n that are 1 or whose prime indices are relatively prime.
- A327529Maximum divisor of n that is 1 or whose prime indices are relatively prime.
- A327528Quotient of n over the maximum uniform divisor of n.
- A327527Number of uniform divisors of n.
- A327526Maximum uniform divisor of n.
- A327525Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.
- A327524Number of factorizations of the n-th uniform number A072774(n) into uniform numbers > 1.
- A327523Number of factorizations of the n-th number with distinct prime multiplicities A130091(n) into numbers > 1 with distinct prime multiplicities.
- A327522Number of factorizations of the n-th prime power A000961(n) into into prime powers > 1.
- A327521Number of factorizations of the n-th squarefree number A005117(n) into squarefree numbers > 1.
- A327520Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.
- A327519Number of factorizations of A305078(n - 1), the n-th connected number, into connected numbers > 1.
- A327518Number of factorizations of A302696(n), the n-th number that is 1, 2, or a nonprime number with pairwise coprime prime indices, into factors > 1 satisfying the same conditions.
- A327517Number of factorizations of n that are empty or have at least two factors, all of which are > 1 and pairwise coprime.
- A327516Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.
- A327515Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1, 2, or a nonprime number whose prime indices are pairwise coprime (A327512, A327514).
- A327514Quotient of n over the maximum divisor of n that is 1, 2, or a nonprime number whose prime indices are pairwise coprime.
- A327513Number of divisors of n that are 1, 2, or a nonprime number whose prime indices are pairwise coprime.
- A327512Maximum divisor of n that is 1, 2, or a nonprime number whose prime indices are pairwise coprime.
- A327503Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).
- A327502Quotient of n over the maximum divisor of n that is 1 or not a perfect power.
- A327501Maximum divisor of n that is 1 or not a perfect power.
- A327500Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor whose prime multiplicities are distinct (A327498, A327499).
- A327499Quotient of n over the maximum divisor of n whose prime multiplicities are distinct.
- A327498Maximum divisor of n whose prime multiplicities are distinct (A130091).
- A327486Product of omegas of positive integers from 2 to n.
- A327485Product of means of the integer partitions with Heinz numbers from 2 to n.
- A327484Number of integer partitions of 2^n whose mean is a power of 2.
- A327483Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.
- A327482Irregular triangle read by rows where T(n,d) is the number of integer partitions of n with mean d|n.
- A327481Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.
- A327478Numbers whose average binary index is also a binary index.
- A327477Number of subsets of {1..n} containing n whose mean is not an element.
- A327476Heinz numbers of integer partitions whose mean A326567/A326568 is not a part.
- A327475Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.
- A327474Number of distinct means of subsets of {1..n}, where {} has mean 0.
- A327473Heinz numbers of integer partitions whose mean (A326567/A326568) is a part.
- A327472Number of integer partitions of n not containing their mean.
- A327471Number of subsets of {1..n} not containing their mean.
- A327438Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.
- A327437Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).
- A327436Number of connected, unlabeled antichains of nonempty subsets covering n vertices with at least one (non-endpoint) cut-vertex.
- A327426Number of non-connected, unlabeled, antichain covers of {1..n} (vertex-connectivity 0).
- A327425Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).
- A327424Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.
- A327407Number of steps to reach a fixed point starting with n and repeatedly taking the quotient over the maximum divisor that is 1, prime, or whose prime indices are pairwise coprime. (A327389, A327401).
- A327406Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).
- A327405Quotient of n over the maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.
- A327404Quotient of n over the maximum divisor of n that is 2 or whose prime indices have a common divisor > 1.
- A327403Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum stable divisor (A327393, A327402).
- A327402Quotient of n over the maximum stable divisor of n.
- A327401Quotient of n over the maximum divisor of n that is 1, prime, or whose prime indices are pairwise coprime.
- A327400Number of factorizations of n that are constant or whose factors are relatively prime.
- A327399Number of factorizations of n that are constant or whose distinct parts are pairwise coprime.
- A327398Maximum connected squarefree divisor of n.
- A327395Quotient of n over the maximum connected divisor of n.
- A327394Number of stable divisors of n.
- A327393Maximum stable divisor of n.
- A327392Irregular triangle read by rows giving the connected components of the prime indices of n.
- A327391Number of divisors of n that are 1, prime, or whose prime indices are pairwise coprime.
- A327390Number of connected divisors of n.
- A327389Maximum divisor of n that is prime or whose prime indices are pairwise coprime.
- A327379Number of labeled non-mating-type graphs with n vertices.
- A327377Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).
- A327376BII-numbers of set-systems with vertex-connectivity 3.
- A327375Number of set-systems with n vertices and vertex-connectivity 2.
- A327374BII-numbers of set-systems with vertex-connectivity 2.
- A327373BII-numbers of complete simple graphs.
- A327372Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).
- A327371Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).
- A327370Number of labeled simple graphs with n vertices and exactly n - 1 endpoints (vertices of degree 1).
- A327369Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).
- A327368The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.
- A327367Number of labeled simple graphs with n vertices, at least one of which is isolated.
- A327366Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and minimum vertex-degree k.
- A327365Triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with vertex-connectivity >= k.
- A327364Number of labeled simple graphs with n vertices, a connected edge-set, and at least one endpoint (vertex of degree 1).
- A327363Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity >= k.
- A327362Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).
- A327359Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.
- A327358Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.
- A327357Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.
- A327356Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).
- A327355Number of antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).
- A327354Number of disconnected or empty antichains of nonempty subsets of {1..n} (non-spanning edge-connectivity 0).
- A327353Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.
- A327352Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.
- A327351Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.
- A327350Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity >= k.
- A327336Number of labeled simple graphs with vertex-connectivity 1.
- A327335Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.
- A327334Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity k.
- A327237Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices that, if the isolated vertices are removed, have cut-connectivity k.
- A327236Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.
- A327235Number of unlabeled simple graphs with n vertices whose edge-set is not connected.
- A327234Smallest BII-number of a set-system with cut-connectivity n.
- A327231Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).
- A327230Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.
- A327229Number of set-systems covering n vertices with at least one endpoint/leaf.
- A327228Number of set-systems with n vertices and at least one endpoint/leaf.
- A327227Number of labeled simple graphs covering n vertices with at least one endpoint/leaf.
- A327201Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs covering n vertices with non-spanning edge-connectivity k.
- A327200Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.
- A327199Number of labeled simple graphs with n vertices whose edge-set is not connected.
- A327198Number of labeled simple graphs covering n vertices with vertex-connectivity 2.
- A327197Number of set-systems covering n vertices with cut-connectivity 1.
- A327196Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (non-spanning edge-connectivity 1).
- A327149Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.
- A327148Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.
- A327147Smallest BII-number of a set-system with spanning edge-connectivity n.
- A327146Number of labeled simple graphs with n vertices and spanning edge-connectivity 2.
- A327145Number of connected set-systems with n vertices and at least one bridge (spanning edge-connectivity 1).
- A327144Spanning edge-connectivity of the set-system with BII-number n.
- A327130Number of set-systems covering n vertices with spanning edge-connectivity 2.
- A327129Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1).
- A327128Number of set-systems with n vertices whose edge-set has cut-connectivity 1.
- A327127Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices where k is the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph.
- A327126Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.
- A327125Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.
- A327114Number of labeled simple graphs covering n vertices with cut-connectivity 1.
- A327113Number of set-systems covering n vertices with cut-connectivity 2.
- A327112Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.
- A327111BII-numbers of set-systems with spanning edge-connectivity 1.
- A327110BII-numbers of set-systems with spanning edge-connectivity 3.
- A327109BII-numbers of set-systems with spanning edge-connectivity >= 2.
- A327108BII-numbers of set-systems with spanning edge-connectivity 2.
- A327107BII-numbers of set-systems with minimum vertex-degree > 1.
- A327106BII-numbers of set-systems with maximum degree 2.
- A327105BII-numbers of set-systems with minimum degree 1.
- A327104Maximum vertex-degree of the set-system with BII-number n.
- A327103Minimum vertex-degree in the set-system with BII-number n.
- A327102BII-numbers of set-systems with non-spanning edge-connectivity >= 2.
- A327101BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).
- A327100BII-numbers of antichains of sets with cut-connectivity 1.
- A327099BII-numbers of set-systems with non-spanning edge-connectivity 1.
- A327098BII-numbers of set-systems with cut-connectivity 1.
- A327097BII-numbers of set-systems with non-spanning edge-connectivity 2.
- A327082BII-numbers of set-systems with cut-connectivity 2.
- A327081BII-numbers of maximal uniform set-systems covering an initial interval of positive integers.
- A327080BII-numbers of maximal uniform set-systems (or complete hypergraphs).
- A327079Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).
- A327078Binomial transform of A001187 (labeled connected graphs), if we assume A001187(1) = 0.
- A327077Triangle read by rows where T(n,k) is the number of unlabeled simple connected graphs with n vertices and k bridges.
- A327076Maximum divisor of n that is 1 or connected.
- A327075Number of non-connected unlabeled simple graphs covering n vertices.
- A327074Number of unlabeled connected graphs with n vertices and exactly one bridge.
- A327073Number of labeled simple connected graphs with n vertices and exactly one bridge.
- A327072Triangle read by rows where T(n,k) is the number of labeled simple connected graphs with n vertices and exactly k bridges.
- A327071Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.
- A327070Number of non-connected simple labeled graphs covering n vertices.
- A327069Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and spanning edge-connectivity k.
- A327062Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.
- A327061BII-numbers of pairwise intersecting set-systems where every two covered vertices appear together in some edge (cointersecting).
- A327060Number of non-isomorphic weight-n weak antichains of multisets where every two vertices appear together in some edge (cointersecting).
- A327059Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.
- A327058Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.
- A327057Number of antichains covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
- A327053Number of T_0 (costrict) set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).
- A327052Number of T_0 (costrict) set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
- A327051Vertex-connectivity of the set-system with BII-number n.
- A327041a(n) is the number whose binary indices are the union of the set-system with BII-number n.
- A327040Number of set-systems covering n vertices, every two of which appear together in some edge (cointersecting).
- A327039Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
- A327038Number of pairwise intersecting set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
- A327037Number of pairwise intersecting set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).
- A327020Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).
- A327019Number of non-isomorphic set-systems of weight n whose dual is a (strict) antichain.
- A327018Number of non-isomorphic set-systems of weight n whose dual is a weak antichain.
- A327017Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.
- A327016BII-numbers of finite T_0 topologies without their empty set.
- A327012Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.
- A327011Number of unlabeled sets of subsets covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 sets of subsets.
- A326979BII-numbers of T_1 set-systems.
- A326978Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.
- A326977Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a (strict) antichain, also called T_1 integer partitions.
- A326976Number of factorizations of n into factors > 1 such that every prime factor of n is the GCD of some subset of the factors.
- A326975Number of factorizations of n into factors > 1 whose dual is a weak antichain.
- A326974Number of unlabeled set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 set-systems.
- A326973Number of unlabeled set-systems covering n vertices whose dual is a weak antichain.
- A326972Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.
- A326971Number of unlabeled set-systems on n vertices whose dual is a weak antichain.
- A326970Number of set-systems covering n vertices whose dual is a weak antichain.
- A326969Number of sets of subsets of {1..n} whose dual is a weak antichain.
- A326968Number of set-systems on n vertices whose dual is a weak antichain.
- A326967Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.
- A326966BII-numbers of set-systems whose dual is a weak antichain.
- A326965Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.
- A326964Number of connected set-systems covering a subset of {1..n}.
- A326961Number of set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called covering T_1 set-systems.
- A326960Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.
- A326959Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.
- A326951Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.
- A326950Number of T_0 antichains of nonempty subsets of {1..n}.
- A326949Number of unlabeled T_0 sets of subsets of {1..n}.
- A326948Number of connected T_0 set-systems on n vertices.
- A326947BII-numbers of T_0 set-systems.
- A326946Number of unlabeled T_0 set-systems on n vertices.
- A326945Number of T_0 sets of subsets of {1..n} that are closed under intersection.
- A326944Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.
- A326943Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.
- A326942Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.
- A326941Number of T_0 sets of subsets of {1..n}.
- A326940Number of T_0 set-systems on n vertices.
- A326939Number of T_0 sets of subsets of {1..n} that cover all n vertices.
- A326913BII-numbers of set-systems (without {}) closed under union and intersection.
- A326912BII-numbers of pairwise intersecting set-systems with empty intersection.
- A326911BII-numbers of set-systems with empty intersection.
- A326910BII-numbers of pairwise intersecting set-systems.
- A326909Number of sets of subsets of {1...n} closed under union and intersection and covering all of the vertices.
- A326908Number of non-isomorphic sets of subsets of {1...n} that are closed under union and intersection.
- A326907Number of non-isomorphic sets of of subsets of {1...n} that are closed under union and cover all n vertices. First differences of A193675.
- A326906Number of sets of subsets of {1...n} that are closed under union and cover all n vertices.
- A326905BII-numbers of set-systems (without {}) closed under intersection.
- A326904Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.
- A326903Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.
- A326902Number of set-systems (without {}) covering n vertices that are closed under intersection.
- A326901Number of set-systems (without {}) on n vertices that are closed under intersection.
- A326900Number of set-systems on n vertices that are closed under union and intersection.
- A326899Number of unlabeled connectedness systems covering n vertices without singletons.
- A326898Number of unlabeled topologies with up to n points.
- A326883Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.
- A326882Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.
- A326881Number of set-systems with {} that are closed under intersection and cover n vertices.
- A326880BII-numbers of set-systems that are closed under nonempty intersection.
- A326879BII-numbers of connected connectedness systems.
- A326878Number of topologies whose points are a subset of {1...n}.
- A326877Number of connectedness systems covering n vertices without singletons.
- A326876BII-numbers of finite topologies without their empty set.
- A326875BII-numbers of set-systems that are closed under union.
- A326874BII-numbers of abstract simplicial complexes.
- A326873BII-numbers of connectedness systems without singletons.
- A326872BII-numbers of connectedness systems.
- A326871Number of unlabeled connectedness systems covering n vertices.
- A326870Number of connectedness systems covering n vertices.
- A326869Number of unlabeled connected connectedness systems on n vertices.
- A326868Number of connected connectedness systems on n vertices.
- A326867Number of unlabeled connectedness systems on n vertices.
- A326866Number of connectedness systems on n vertices.
- A326854BII-numbers of T_0 (costrict), pairwise intersecting set-systems where every two vertices appear together in some edge (cointersecting).
- A326853BII-numbers of set-systems where every two covered vertices appear together in some edge (cointersecting).
- A326852Number of non-constant integer partitions of n whose length and maximum both divide n.
- A326851Number of strict integer partitions of n whose length and maximum both divide n.
- A326850Number of strict integer partitions of n whose maximum part divides n.
- A326849Number of integer partitions of n whose length times maximum is a multiple of n.
- A326848Heinz numbers of integer partitions of m >= 0 whose length times maximum is a multiple of m.
- A326847Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.
- A326846Length times maximum of the integer partition with Heinz number n.
- A326845Sum times maximum of the integer partition with Heinz number n.
- A326844Let y be the integer partition with Heinz number n. Then a(n) is the size of the complement, in the minimal rectangular partition containing the Young diagram of y, of the Young diagram of y.
- A326843Number of integer partitions of n whose length and maximum both divide n.
- A326842Number of integer partitions of n whose parts all divide n and whose length also divides n.
- A326841Heinz numbers of integer partitions of m >= 0 using divisors of m.
- A326840Denominator of A056239(n)/A061395(n).
- A326839Numerator of A056239(n)/A061395(n) where A056239 is sum of prime indices and A061395 is maximum prime index.
- A326838Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.
- A326837Heinz numbers of integer partitions whose length and maximum both divide their sum.
- A326836Heinz numbers of integer partitions whose maximum part divides their sum.
- A326788BII-numbers of simple labeled graphs.
- A326787Edge-connectivity of the set-system with BII-number n.
- A326786Cut-connectivity of the set-system with BII-number n.
- A326785BII-numbers of uniform regular set-systems.
- A326784BII-numbers of regular set-systems.
- A326783BII-numbers of uniform set-systems.
- A326782Numbers whose binary indices are prime numbers.
- A326781No position of a 1 in the reversed binary digits of n is a power of 2.
- A326754BII-numbers of set-systems covering an initial interval of positive integers.
- A326753Number of connected components of the set-system with BII-number n.
- A326752BII-numbers of hypertrees.
- A326751BII-numbers of blobs.
- A326750BII-numbers of clutters (connected antichains of nonempty sets).
- A326749BII-numbers of connected set-systems.
- A326704BII-numbers of antichains of nonempty sets.
- A326703BII-numbers of chains of nonempty sets.
- A326702Number of distinct vertices in the set-system with BII-number n.
- A326701BII-numbers of set partitions.
- A326700Denominator of the average position of a 1 in the reversed binary digits of n.
- A326699Numerator of the average position of a 1 in the reversed binary digits of n.
- A326675The positions of 1's in the reversed binary expansion of n are pairwise coprime, where a singleton is not coprime unless it is {1}.
- A326674GCD of the set of positions of 1's in the reversed binary digits of n.
- A326673The positions of ones in the reversed binary digits of n have integer geometric mean.
- A326672The positions of ones in the binary digits of n have integer geometric mean.
- A326671Number of factorizations of 2^n into factors > 1 with even integer average.
- A326670Number of strict integer partitions y of n such that the average of the set {2^(s - 1): s in y} is an integer.
- A326669Numbers n such that the average position of a one in the binary digits of n is an integer.
- A326668Number of strict factorizations of 2^n into factors > 1 with integer average.
- A326667Number of factorizations of 2^n into factors > 1 with integer average.
- A326666Numbers n such that there exists a factorization of n into factors > 1 whose mean is not an integer but whose geometric mean is an integer.
- A326647Number of factorizations of n into factors > 1 with integer average and integer geometric mean.
- A326646Heinz numbers of non-constant integer partitions whose mean and geometric mean are both integers.
- A326645Heinz numbers of integer partitions whose mean and geometric mean are both integers.
- A326644Number of subsets of {1...n} containing n whose mean and geometric mean are both integers.
- A326643Number of subsets of {1...n} whose mean and geometric mean are both integers.
- A326642Number of non-constant integer partitions of n whose mean and geometric mean are both integers.
- A326641Number of integer partitions of n whose mean and geometric mean are both integers.
- A326625Number of strict integer partitions of n whose geometric mean is an integer.
- A326624Heinz numbers of non-constant integer partitions whose geometric mean is an integer.
- A326623Heinz numbers of integer partitions whose geometric mean is an integer.
- A326622Number of factorizations of n into factors > 1 with integer average.
- A326621Numbers n such that the average of the set of distinct prime indices of n is an integer.
- A326620Denominator of the average of the set of distinct prime indices of n.
- A326619Numerator of the average of the set of distinct prime indices of n.
- A326574Number of antichains of subsets of {1...n} with equal edge-sums.
- A326573Number of connected antichains of subsets of {1...n}, all having different sums.
- A326572Number of covering antichains of subsets of {1...n}, all having different sums.
- A326571Number of covering antichains of nonempty, non-singleton subsets of {1...n}, all having different sums.
- A326570Number of covering antichains of subsets of {1...n} with different edge-sizes.
- A326569Number of covering antichains of subsets of {1...n} with no singletons and different edge-sizes.
- A326568Denominator of the average of the multiset of prime indices of n.
- A326567Numerator of the average of the multiset of prime indices of n.
- A326566Number of covering antichains of subsets of {1...n} with equal edge-sums.
- A326565Number of covering antichains of nonempty, non-singleton subsets of {1...n}, all having the same sum.
- A326537MM-numbers of multiset partitions where each part has a different average.
- A326536MM-numbers of multiset partitions where every part has the same average.
- A326535MM-numbers of multiset partitions where each part has a different sum.
- A326534MM-numbers of multiset partitions where every part has the same sum.
- A326533MM-numbers of multiset partitions where each part has a different length.
- A326521Number of normal multiset partitions of weight n where each part has a different average.
- A326520Number of normal multiset partitions of weight n where every part has the same average.
- A326519Number of normal multiset partitions of weight n where each part has a different sum.
- A326518Number of normal multiset partitions of weight n where every part has the same sum.
- A326517Number of normal multiset partitions of weight n where each part has a different size.
- A326516Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.
- A326515Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.
- A326514Number of factorizations of n into factors > 1 where each factor has a different number of prime factors counted with multiplicity.
- A326513Number of set partitions of {1...n} where each block has a different average.
- A326512Number of set partitions of {1...n} where every block has the same average.
- A326498Number of maximal subsets of {1...n} containing no sums of distinct elements.
- A326497Number of maximal sum-free and product-free subsets of {1...n}.
- A326496Number of maximal product-free subsets of {1...n}.
- A326495Number of subsets of {1...n} containing no sums or products of pairs of elements.
- A326494Number of subsets of {1...n} containing all differences and quotients of pairs of distinct elements.
- A326493Number of subsets of {1...n} containing no quotients of divisible elements.
- A326492Number of maximal subsets of {1...n} containing no quotients of pairs of distinct elements.
- A326491Number of maximal subsets of {1...n} containing no differences or quotients of pairs of distinct elements.
- A326490Number of subsets of {1...n} containing no differences or quotients of pairs of distinct elements.
- A326489Number of product-free subsets of {1...n}.
- A326442Number of subsets of {1..n} whose sum is one fewer than the product of their complement.
- A326441Number of subsets of {1...n} whose sum is equal to the product of their complement.
- A326439Number of maximal subsets of {1...n} such that no two elements have the same sorted prime signature.
- A326438Number of subsets of {1...n} such that no two elements have the same sorted prime signature.
- A326375Number of intersecting antichains of subsets of {1...n} with empty intersection (meaning there is no vertex in common to all the edges).
- A326374Irregular triangle read by rows where T(n,d) for d|n is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices.
- A326373Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.
- A326372Number of intersecting antichains of (possibly empty) subsets of {1...n}.
- A326366Number of intersecting antichains of nonempty subsets of {1...n} with empty intersection (meaning there is no vertex in common to all the edges).
- A326365Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.
- A326364Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.
- A326363Number of maximal intersecting antichains of subsets of {1...n}.
- A326362Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1...n}.
- A326361Number of maximal intersecting antichains of sets covering n vertices with no singletons.
- A326360Number of maximal antichains of nonempty, non-singleton subsets of {1...n}.
- A326359Number of maximal antichains of nonempty subsets of {1...n}.
- A326358Number of maximal antichains of subsets of {1...n}.
- A326351Number of non-nesting connected simple graphs on a subset of {1...n}.
- A326350Number of non-nesting connected simple graphs with vertices {1...n}.
- A326349Number of non-nesting, topologically connected simple graphs covering {1...n}.
- A326341Number of minimal topologically connected chord graphs covering {1...n}.
- A326340Number of maximal simple graphs with vertices {1...n} and no crossing or nesting edges.
- A326339Number of connected simple graphs with vertices {1...n} and no crossing or nesting edges.
- A326338Number of simple graphs with vertices {1...n} whose weakly nesting edges are connected.
- A326337Number of simple graphs covering the vertices {1...n} whose weakly nesting edges are connected.
- A326336Number of set partitions of {1...n} whose capturing blocks are connected.
- A326335Number of set partitions of {1...n} whose nesting blocks are connected.
- A326334Number of sortable factorizations of n.
- A326333Number of integer partitions of n with sortable prime factors.
- A326332Number of integer partitions of n with unsortable prime factors.
- A326331Number of simple graphs covering the vertices {1...n} whose nesting edges are connected.
- A326330Number of simple graphs with vertices {1...n} whose nesting edges are connected.
- A326329Number of simple graphs covering {1...n} with no crossing or nesting edges.
- A326294Number of connected simple graphs on a subset of {1...n} with no crossing or nesting edges.
- A326293Number of non-nesting, topologically connected simple graphs with vertices {1...n}.
- A326292Number of crossing integer partitions of n.
- A326291Number of unsortable factorizations of n.
- A326290Number of non-crossing n-vertex graphs with loops.
- A326289a(0) = 0, a(n) = 2^binomial(n,2) - 2^(n - 1).
- A326279Number of labeled n-vertex simple graphs containing either a crossing or a nesting pair of edges.
- A326278Number of n-vertex, 2-edge multigraphs that are not nesting. Number of n-vertex, 2-edge multigraphs that are not crossing.
- A326277Number of crossing normal multiset partitions of weight n.
- A326260MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).
- A326259MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).
- A326258MM-numbers of unsortable multiset partitions (with empty parts allowed).
- A326257MM-numbers of weakly nesting multiset partitions.
- A326256MM-numbers of nesting multiset partitions.
- A326255MM-numbers of capturing multiset partitions.
- A326254Number of non-capturing set partitions of {1...n}.
- A326253Number of sequences of distinct ordered pairs of positive integers up to n.
- A326252Number of digraphs with vertices {1...n} whose increasing edges are crossing.
- A326251Number of digraphs with vertices {1...n} whose increasing edges are not crossing.
- A326250Number of weakly nesting simple graphs with vertices {1...n}.
- A326249Number of capturing set partitions of {1...n} that are not nesting.
- A326248Number of crossing, nesting set partitions of {1...n}.
- A326247Number of labeled n-vertex 2-edge multigraphs that are neither crossing nor nesting.
- A326246Number of crossing, capturing set partitions of {1...n}.
- A326245Number of crossing, non-capturing set partitions of {1...n}.
- A326244Number of labeled n-vertex simple graphs without crossing or nesting edges.
- A326243Number of capturing set partitions of {1...n}.
- A326240Number of Hamiltonian labeled n-vertex graphs with loops.
- A326239Number of non-Hamiltonian labeled n-vertex graphs with loops.
- A326237Number of non-nesting digraphs with vertices {1...n}, where two edges (a,b), (c,d) are nesting if a < c and b > d or a > c and b < d.
- A326226Number of unlabeled n-vertex Hamiltonian digraphs (with loops).
- A326225Number of Hamiltonian unlabeled n-vertex digraphs (without loops).
- A326224Number of unlabeled n-vertex digraphs (with loops) not containing a Hamiltonian path.
- A326223Number of non-Hamiltonian unlabeled n-vertex digraphs (with loops).
- A326222Number of non-Hamiltonian unlabeled n-vertex digraphs (without loops).
- A326221Number of unlabeled n-vertex digraphs (with loops) containing a Hamiltonian path.
- A326220Number of non-Hamiltonian labeled n-vertex digraphs (with loops).
- A326219Number of labeled n-vertex Hamiltonian digraphs (without loops).
- A326218Number of non-Hamiltonian labeled n-vertex digraphs (without loops).
- A326217Number of labeled n-vertex digraphs (without loops) containing a Hamiltonian path.
- A326216Number of labeled n-vertex digraphs (without loops) not containing a (directed) Hamiltonian path.
- A326215Number of Hamiltonian unlabeled n-vertex graphs with loops.
- A326214Number of labeled n-vertex digraphs (with loops) containing a (directed) Hamiltonian path.
- A326213Number of labeled n-vertex digraphs (with loops) not containing a (directed) Hamiltonian path.
- A326212Number of sortable normal multiset partitions of weight n.
- A326211Number of unsortable normal multiset partitions of weight n.
- A326210Number of labeled simple graphs with vertices {1...n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.
- A326209Number of nesting labeled digraphs with with vertices {1...n}.
- A326208Number of Hamiltonian labeled simple graphs with n vertices.
- A326207Number of non-Hamiltonian labeled simple graphs with n vertices.
- A326206Number of n-vertex labeled simple graphs containing a Hamiltonian path.
- A326205Number of n-vertex labeled simple graphs not containing a Hamiltonian path.
- A326204Number of Hamiltonian labeled n-vertex digraphs (with loops).
- A326180Number of maximal subsets of {1...n} containing n whose product is divisible by their sum.
- A326179Number of subsets of {1...n} containing n whose product is divisible by their sum.
- A326178Number of subsets of {1...n} whose product is equal to their sum.
- A326175Number of minimal subsets of {1...n} containing n whose sum is greater than or equal to the sum of their complement.
- A326174Number of subsets of {1...n} containing n whose sum is greater than or equal to the sum of their complement.
- A326173Number of maximal subsets of {1...n} whose sum is less than or equal to the sum of their complement.
- A326172Number of nonempty subsets of {2...n} whose product is divisible by their sum.
- A326158Nonprime squarefree numbers whose product of prime indices is divisible by their sum of prime indices.
- A326157Squarefree numbers whose product of prime indices is twice their sum of prime indices.
- A326156Number of nonempty subsets of {1...n} whose product is divisible by their sum.
- A326155Positive integers whose sum of prime indices is divisible by their product of prime indices.
- A326154Denominator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.
- A326153Numerator of the product of prime indices of n > 1 divided by the sum of prime indices of n, n > 1.
- A326152Number of integer partitions of n whose product of parts is 2 * n.
- A326151Numbers whose product of prime indices is twice their sum of prime indices.
- A326150Nonprime numbers whose product of prime indices is divisible by their sum of prime indices.
- A326149Numbers whose product of prime indices is divisible by their sum of prime indices.
- A326117Number of subsets of {1...n} containing no products of two or more distinct elements.
- A326116Number of subsets of {2...n} containing no products of two or more distinct elements.
- A326115Number of maximal double-free subsets of {1...n}.
- A326114Number of subsets of {2..n} containing no product of two or more (not necessarily distinct) elements.
- A326083Number of subsets of {1...n} containing all of their pairwise sums <= n.
- A326082Number of maximal sets of pairwise indivisible divisors of n.
- A326081Number of subsets of {1...n} containing the product of any set of distinct elements whose product is <= n.
- A326080Number of subsets of {1...n} containing the sum of every subset whose sum is <= n.
- A326079Number of subsets of {1...n} containing all of their integer quotients > 1.
- A326078Number of subsets of {2...n} containing all of their integer quotients > 1.
- A326077Number of maximal pairwise indivisible subsets of {1...n}.
- A326076Number of subsets of {1...n} containing all of their integer products <= n.
- A326037Heinz numbers of uniform perfect integer partitions.
- A326036Number of uniform complete integer partitions of n.
- A326035Number of uniform knapsack partitions of n.
- A326034Number knapsack partitions of n with largest part 3.
- A326033Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.
- A326032a(2^x + ... + 2^z) = w(x) + ... + w(z), where x...z are distinct nonnegative integers and w = A000120.
- A326031Weight of the set-system with BII-number n.
- A326030Number of antichains of subsets of {1...n} with different edge-sums.
- A326029Number of strict integer partitions of n whose mean and geometric mean are both integers.
- A326028Number of factorizations of n into factors > 1 with integer geometric mean.
- A326027Number of subsets of {1...n} whose geometric mean is an integer.
- A326026Number of non-isomorphic multiset partitions of weight n where each part has a different length.
- A326025Number of maximal subsets of {1...n} containing no sums or products of distinct elements.
- A326024Number of subsets of {1...n} containing no sums or products of distinct elements.
- A326023Number of subsets of {1...n} containing all of their integer quotients.
- A326022Number of minimal complete subsets of {1...n} with maximum n.
- A326021Number of complete subsets of {1...n} with maximum n.
- A326020Number of complete subsets of {1...n}.
- A326019Heinz numbers of non-knapsack partitions such that every non-singleton submultiset has a different sum.
- A326018Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.
- A326017Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.
- A326016Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.
- A326015Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.
- A325994Heinz numbers of integer partitions such that not every ordered pair of distinct parts has a different quotient.
- A325993Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different product.
- A325992Heinz numbers of integer partitions such that not every ordered pair of distinct parts has a different difference.
- A325991Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.
- A325990Numbers with more than one perfect factorization.
- A325989Number of perfect factorizations of n.
- A325988Number of covering (or complete) factorizations of n.
- A325987Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.
- A325986Heinz numbers of complete strict integer partitions.
- A325880Number of maximal subsets of {1...n} containing n such that every ordered pair of distinct elements has a different difference.
- A325879Number of maximal subsets of {1...n} such that every ordered pair of distinct elements has a different difference.
- A325878Number of maximal subsets of {1...n} such that every orderless pair of distinct elements has a different sum.
- A325877Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.
- A325876Number of strict Golomb partitions of n.
- A325875Number of compositions of n whose differences of all degrees > 1 are nonzero.
- A325874Number of integer partitions of n whose differences of all degrees > 1 are nonzero.
- A325869Number of maximal subsets of {1...n} containing n such that every pair of distinct elements has a different quotient.
- A325868Number of subsets of {1...n} containing n such that every ordered pair of distinct elements has a different quotient.
- A325867Number of maximal subsets of {1...n} containing n such that every subset has a different sum.
- A325866Number of subsets of {1...n} containing n such that every subset has a different sum.
- A325865Number of maximal subsets of {1...n} of which every subset has a different sum.
- A325864Number of subsets of {1...n} of which every subset has a different sum.
- A325863Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum.
- A325861Number of maximal subsets of {1...n} such that every pair of distinct elements has a different quotient.
- A325860Number of subsets of {1...n} such that every pair of distinct elements has a different quotient.
- A325859Number of maximal subsets of {1...n} such that every orderless pair of distinct elements has a different product.
- A325858Number of Golomb partitions of n.
- A325857Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.
- A325856Number of integer partitions of n such that every pair of distinct parts has a different product.
- A325855Number of strict integer partitions of n such that every pair of distinct parts has a different product.
- A325854Number of strict integer partitions of n such that every pair of distinct parts has a different quotient.
- A325853Number of integer partitions of n such that every pair of distinct parts has a different quotient.
- A325852Number of (strict) integer partitions of n whose differences of all degrees are nonzero.
- A325851Number of (strict) compositions of n whose differences of all degrees are nonzero.
- A325850Number of permutations of {1...n} whose differences of all degrees are nonzero.
- A325849Number of strict compositions of n with no three consecutive parts in arithmetic progression.
- A325836Number of integer partitions of n having n - 1 different submultisets.
- A325835Number of integer partitions of 2*n having one more distinct submultiset than distinct subset-sums.
- A325834Number of integer partitions of n whose number of submultisets is less than or equal to n.
- A325833Number of integer partitions of n whose number of submultisets is less than n.
- A325832Number of integer partitions of n whose number of submultisets is greater than or equal to n.
- A325831Number of integer partitions of n whose number of submultisets is greater than n.
- A325830Number of integer partitions of 2*n having exactly 2*n submultisets.
- A325828Number of integer partitions of n having exactly n + 1 submultisets.
- A325802Numbers with one more divisor than distinct subset-sums of their prime indices.
- A325801Number of divisors of n minus the number of distinct positive subset-sums of the prime indices of n.
- A325800Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.
- A325799Sum of the prime indices of n minus the number of distinct positive subset-sums of the prime indices of n.
- A325798Numbers with at most as many divisors as the sum of their prime indices.
- A325797Numbers with fewer divisors than the sum of their prime indices.
- A325796Numbers with at least as many divisors as the sum of their prime indices.
- A325795Numbers with more divisors than the sum of their prime indices.
- A325794Number of divisors of n minus the sum of prime indices of n.
- A325793Positive integers whose number of divisors is equal to their sum of prime indices.
- A325792Positive integers with as many proper divisors as the sum of their prime indices.
- A325791Number of necklace permutations of {1...n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.
- A325790Number of permutations of {1...n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.
- A325789Number of perfect necklace compositions of n.
- A325788Number of complete strict necklace compositions of n.
- A325787Number of perfect strict necklace compositions of n.
- A325786Number of complete necklace compositions of n.
- A325782Heinz numbers of strict perfect integer partitions.
- A325781Heinz numbers of complete integer partitions.
- A325780Heinz numbers of perfect integer partitions.
- A325779Heinz numbers of integer partitions for which every restriction to a subinterval has a different sum.
- A325778Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.
- A325777Heinz numbers of integer partitions whose distinct consecutive subsequences do not have different sums.
- A325770Number of distinct consecutive subsequence-sums of the integer partition with Heinz number n.
- A325769Number of integer partitions of n whose distinct consecutive subsequences have different sums.
- A325768Number of integer partitions of n for which every restriction to a subinterval has a different sum.
- A325767Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).
- A325766Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).
- A325765Number of integer partitions of n with a unique consecutive subsequence summing to every positive integer from 1 to n.
- A325764Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.
- A325763Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.
- A325762Heinz numbers of integer partitions with no part greater than the number of ones.
- A325761Heinz numbers of integer partitions whose length is itself a part.
- A325760Heinz number of the frequency span of n.
- A325759Number of distinct frequencies in the frequency span of n.
- A325758Irregular triangle read by rows giving the frequency span signature of n.
- A325757Irregular triangle read by rows giving the frequency span of n.
- A325756A number n belongs to the sequence if n = 1 or n is divisible by its prime shadow A181819(n) and the quotient n/A181819(n) also belongs to the sequence.
- A325755Numbers n divisible by their prime shadow A181819(n).
- A325710Number of maximal subsets of {1...n} containing no products of distinct elements.
- A325709Replace k with k! in the prime indices of n.
- A325708Numbers n whose prime indices cover an initial interval of positive integers and include all prime exponents of n.
- A325707Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.
- A325706Heinz numbers of integer partitions containing all of their distinct multiplicities.
- A325705Number of integer partitions of n containing all of their distinct multiplicities.
- A325704If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.
- A325703If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.
- A325702Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).
- A325701Nonprime Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.
- A325700Numbers with as many distinct even as distinct odd prime indices.
- A325699Number of distinct even prime indices of n minus the number of distinct odd prime indices of n.
- A325698Numbers with as many even as odd prime indices, counted with multiplicity.
- A325697Number of rooted trees with n vertices with no proper terminal subtree appearing at only one position.
- A325696Number of length-3 strict compositions of n such that no part is the sum of the other two.
- A325695Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.
- A325694Numbers with one fewer divisors than the sum of their prime indices.
- A325691Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.
- A325690Number of length-3 integer partitions of n whose largest part is not the sum of the other two.
- A325689Number of length-3 compositions of n such that no part is the sum of the other two.
- A325688Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.
- A325687Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.
- A325686Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.
- A325685Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.
- A325684Number of minimal complete rulers of length n.
- A325683Number of maximal Golomb rulers of length n.
- A325682Number of necklace compositions of n such that every distinct circular subsequence has a different sum.
- A325681Number of necklace compositions of n such that every restriction to a circular subinterval has a different sum.
- A325680Number of compositions of n such that every distinct circular subsequence has a different sum.
- A325679Number of compositions of n such that every restriction to a circular subinterval has a different sum.
- A325678Maximum length of a composition of n such that every restriction to a subinterval has a different sum.
- A325677Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.
- A325676Number of compositions of n such that every distinct consecutive subsequence has a different sum.
- A325663Matula-Goebel numbers of not necessarily regular rooted stars.
- A325662Matula-Goebel numbers of regular rooted stars.
- A325661q-powerful numbers. Numbers whose factorization into factors prime(i)/i has no factor of multiplicity 1.
- A325660Number of ones in the q-signature of n.
- A325625Sorted prime signature of 2^n - 1.
- A325624a(n) = prime(n)^(n!).
- A325623Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.
- A325622Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.
- A325621Heinz numbers of integer partitions whose reciprocal factorial sum is an integer.
- A325620Number of integer partitions of n whose reciprocal factorial sum is an integer.
- A325619Heinz numbers of integer partitions whose reciprocal factorial sum is 1.
- A325618Numbers n such that there exists an integer partition of n whose reciprocal factorial sum is 1.
- A325617Multinomial coefficient of the prime signature of n!.
- A325616Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.
- A325615Sorted q-signature of n.
- A325614Unsorted q-signature of n.
- A325613Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.
- A325612Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.
- A325611Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.
- A325610Adjusted frequency depth of 2^n - 1.
- A325609Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i.
- A325608Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.
- A325592Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.
- A325591Number of compositions of n with circular differences all equal to 1, 0, or -1.
- A325590Number of necklace compositions of n with circular differences all equal to 1 or -1.
- A325589Number of compositions of n whose circular differences are all 1 or -1.
- A325588Number of necklace compositions of n with equal circular differences up to sign.
- A325558Number of compositions of n with equal circular differences up to sign.
- A325557Number of compositions of n with equal differences up to sign.
- A325556Number of necklace compositions of n with distinct circular differences up to sign.
- A325555Number of necklace compositions of n with distinct differences up to sign.
- A325554Number of necklace compositions of n with distinct differences.
- A325553Number of compositions of n with distinct circular differences up to sign.
- A325552Number of compositions of n with distinct differences up to sign.
- A325551Number of compositions of n with distinct circular differences.
- A325550Number of necklace compositions of n with distinct multiplicities.
- A325549Number of necklace compositions of n with distinct circular differences.
- A325548Number of compositions of n with strictly decreasing differences.
- A325547Number of compositions of n with strictly increasing differences.
- A325546Number of compositions of n with weakly increasing differences.
- A325545Number of compositions of n with distinct differences.
- A325544Number of nodes in the rooted tree with Matula-Goebel number n!.
- A325543Width (number of leaves) of the rooted tree with Matula-Goebel number n!.
- A325538Number of subsets of {1...n} whose product is one more than the sum of their complement.
- A325537Irregular triangle whose rows are the sorted combined parts of all strict integer partitions of n.
- A325536Sum of sums of omegas of parts over all integer partitions of n.
- A325515Sum of sums of omegas of the parts over all strict integer partitions of n.
- A325514Heinz number of row n of the triangle of partition numbers A008284.
- A325513Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.
- A325512Number of distinct nonzero numbers of partitions of n counted by length.
- A325511Numbers whose prime signature is that of a factorial number.
- A325510Number of non-isomorphic multiset partitions of the multiset of prime indices of n!.
- A325509Number of factorizations of n! into factorial numbers.
- A325508Product of primes indexed by the prime exponents of n!.
- A325507Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all integer partitions of n.
- A325506Product of Heinz numbers over all strict integer partitions of n.
- A325505Heinz number of the set of Heinz numbers of all strict integer partitions of n.
- A325504Product of products of parts over all strict integer partitions of n.
- A325503Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.
- A325502Heinz number of row n of Pascal's triangle A007318.
- A325501Product of Heinz numbers over all integer partitions of n.
- A325500Heinz number of the set of Heinz numbers of integer partitions of n. Heinz numbers of rows of A215366.
- A325468Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.
- A325467Heinz numbers of integer partitions y such that the k-th differences of y are distinct (independently) for all k >= 0.
- A325466Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.
- A325461Heinz numbers of integer partitions with strictly decreasing differences (with the last part taken to be 0).
- A325460Heinz numbers of integer partitions with strictly increasing differences (with the last part taken to be 0).
- A325459Number of integer partitions of n that are not hooks but whose augmented differences are hooks.
- A325458Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.
- A325457Heinz numbers of integer partitions with strictly decreasing differences.
- A325456Heinz numbers of integer partitions with strictly increasing differences.
- A325416Least k such that the omega-sequence of k sums to n, and 0 if none exists.
- A325415Number of distinct sums of omega-sequences of integer partitions of n.
- A325414Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.
- A325413Largest sum of the omega-sequence of an integer partition of n.
- A325412Number of distinct omega-sequences of integer partitions of n.
- A325411Numbers whose omega-sequence has repeated parts.
- A325410Smallest k such that the adjusted frequency depth of k! is n > 2.
- A325407Nonprime Heinz numbers of multiples of triangular partitions, or of finite arithmetic progressions with offset 0.
- A325406Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.
- A325405Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the the i-th differences for i != k.
- A325404Number of reversed integer partitions y of n such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the the i-th differences for i != k.
- A325403Number of permutations of the multiset of prime factors of 2n whose first part is not 2.
- A325400Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.
- A325399Heinz numbers of integer partitions whose k-th differences are strictly decreasing for all k >= 0.
- A325398Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.
- A325397Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.
- A325396Heinz numbers of integer partitions whose augmented differences are strictly decreasing.
- A325395Heinz numbers of integer partitions whose augmented differences are strictly increasing.
- A325394Heinz numbers of integer partitions whose augmented differences are weakly increasing.
- A325393Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.
- A325392Number of permutations of the multiset of prime factors of n whose first part is not 2.
- A325391Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.
- A325390Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).
- A325389Heinz numbers of integer partitions whose augmented differences are weakly decreasing.
- A325388Heinz numbers of strict integer partitions with distinct differences (with the last part taken to be 0).
- A325387Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.
- A325374Numbers with adjusted frequency depth 3 whose prime indices cover an initial interval of positive integers.
- A325373Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.
- A325372Totally abnormal numbers. Heinz numbers of totally abnormal integer partitions.
- A325371Numbers whose prime signature has multiplicities of its parts all distinct and covering an initial interval of positive integers.
- A325370Numbers whose prime signature has multiplicities covering an initial interval of positive integers.
- A325369Numbers with no two prime exponents appearing the same number of times in the prime signature.
- A325368Heinz numbers of integer partitions with distinct differences between successive parts.
- A325367Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).
- A325366Heinz numbers of integer partitions whose whose augmented differences are distinct.
- A325365Number of maximal subsets of {1...n} containing n such that no two elements have the same sorted prime signature.
- A325364Heinz numbers of integer partitions whose differences (with the last part taken to be zero) are weakly decreasing.
- A325363Heinz numbers of integer partitions into nonzero triangular numbers A000217.
- A325362Heinz numbers of integer partitions whose differences (with the last part taken to be 0) are weakly increasing.
- A325361Heinz numbers of integer partitions whose differences are weakly decreasing.
- A325360Heinz numbers of integer partitions whose differences are weakly increasing.
- A325359Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.
- A325358Number of integer partitions of n whose augmented differences are strictly decreasing.
- A325357Number of integer partitions of n whose augmented differences are strictly increasing.
- A325356Number of integer partitions of n whose augmented differences are weakly increasing.
- A325355One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.
- A325354Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.
- A325353Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.
- A325352Heinz number of the differences plus one of the integer partition with Heinz number n.
- A325351Heinz number of the augmented differences of the integer partition with Heinz number n.
- A325350Number of integer partitions of n whose augmented differences are weakly decreasing.
- A325349Number of integer partitions of n whose augmented differences are distinct.
- A325337Numbers whose prime exponents are distinct and cover an initial interval of positive integers.
- A325336Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k whose parts cover an initial interval of positive integers.
- A325335Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.
- A325334Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.
- A325333Number of integer partitions of n whose multiplicities all appear the same number of times.
- A325332Number of totally abnormal integer partitions of n.
- A325331Number of integer partitions of n whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers.
- A325330Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.
- A325329Number of integer partitions of n whose multiplicities appear with distinct multiplicities.
- A325328Heinz numbers of finite arithmetic progressions (integer partitions with equal differences).
- A325327Heinz numbers of multiples of triangular partitions, or finite arithmetic progressions with offset 0.
- A325326Heinz numbers of integer partitions covering an initial interval of positive integers with distinct multiplicities.
- A325325Number of integer partitions of n with distinct differences between successive parts.
- A325324Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.
- A325285Number of integer partitions of n whose omega-sequence has repeated parts.
- A325284Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.
- A325283Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.
- A325282Maximum adjusted frequency depth among integer partitions of n.
- A325281Numbers of the form a*b, a*a*b, or a*a*b*c where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).
- A325280Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.
- A325279Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.
- A325278Smallest number with adjusted frequency depth n.
- A325277Irregular triangle read by rows where row 1 is {1} and row n is the sequence starting with n and repeatedly applying A181819 until a prime number is reached.
- A325276Irregular triangle read by rows where row n is the omega-sequence of n!.
- A325275Heinz number of the omega-sequence of n!.
- A325274Sum of the omega-sequence of n!.
- A325273Prime omicron of n!.
- A325272Adjusted frequency depth of n!.
- A325271Number of integer partitions of n with frequency depth round(sqrt(n)).
- A325270Numbers with one fewer distinct prime exponents than (not necessarily distinct) prime factors.
- A325269Number of integer partitions of n with 2 distinct parts or at least 3 parts.
- A325268Regular triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.
- A325267Number of integer partitions of n with omicron 2.
- A325266Numbers whose adjusted frequency depth equals their number of prime factors counted with multiplicity.
- A325265Numbers with sum of omega-sequence > 4.
- A325264Numbers whose omega-sequence sums to 7.
- A325263Number of subsets of {1...n} containing n such that no two elements have the same sorted prime signature.
- A325262Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.
- A325261Numbers whose omega-sequence does not cover an initial interval of positive integers.
- A325260Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.
- A325259Numbers with one fewer distinct prime exponents than distinct prime factors.
- A325258a(1) = 1; otherwise, first differences of Levine's sequence A011784.
- A325257Number of strongly normal multisets of size n whose omega-sequence is strict (no repeated parts).
- A325256Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size.
- A325255Maximum frequency depth of normal multisets of size n.
- A325254Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.
- A325253Number of integer partitions of n with adjusted frequency depth ceiling(sqrt(n)).
- A325252Number of integer partitions of n with adjusted frequency depth floor(sqrt(n)).
- A325251Numbers whose omega-sequence covers an initial interval of positive integers.
- A325250Number of integer partitions of n whose omega-sequence is strict (no repeated parts).
- A325249Sum of the omega-sequence of n.
- A325248Heinz number of the omega-sequence of n.
- A325247Numbers whose omega-sequence is strict (no repeated parts).
- A325246Number of integer partitions of n with adjusted frequency depth equal to their length.
- A325245Number of integer partitions of n with adjusted frequency depth 3.
- A325244Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.
- A325243Number of integer partitions of n with exactly two distinct multiplicities.
- A325242Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.
- A325241Numbers > 1 whose maximum prime exponent is one greater than their minimum.
- A325240Numbers whose minimum prime exponent is 2.
- A325239Irregular triangle read by rows where row 1 is {1} and row n > 1 is the sequence starting with n and repeatedly applying A181819 until 2 is reached.
- A325238First positive integer with each omega-sequence.
- A325235Heinz numbers of integer partitions with Dyson rank 1 or -1.
- A325234Heinz numbers of integer partitions with Dyson rank -1.
- A325233Heinz numbers of integer partitions with Dyson rank 1.
- A325232Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.
- A325231Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.
- A325230Numbers of the form p^k * q, p and q prime, p > q, k > 0.
- A325229Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.
- A325228Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.
- A325227Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.
- A325226Number of prime factors of n that are less than the largest, counted with multiplicity.
- A325225Lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.
- A325224Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.
- A325223Sum of the prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n.
- A325200Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.
- A325199Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.
- A325198Positive integers whose maximum prime index minus minimum prime index is 2.
- A325197Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.
- A325196Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.
- A325195Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.
- A325194Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.
- A325193Number of integer partitions whose sum plus co-rank is n, where co-rank is maximum of length and largest part.
- A325192Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.
- A325191Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.
- A325190Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.
- A325189Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.
- A325188Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.
- A325187Number of integer partitions of n such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.
- A325186Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.
- A325185Heinz numbers of integer partitions such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.
- A325184Last part of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.
- A325183Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.
- A325182Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.
- A325181Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.
- A325180Heinz number of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.
- A325179Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.
- A325178Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.
- A325170Heinz numbers of integer partitions with origin-to-boundary graph-distance equal to 2.
- A325169Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.
- A325168Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.
- A325167Heinz number of the internal portion of the integer partition with Heinz number n.
- A325166Size of the internal portion of the integer partition with Heinz number n.
- A325165Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.
- A325164Heinz numbers of integer partitions with Durfee square of length 2.
- A325163Heinz number of the inner lining partition of the integer partition with Heinz number n.
- A325162Squarefree numbers with no two prime indices differing by less than 3.
- A325161Nonprime squarefree numbers not divisible by any two consecutive primes.
- A325160Products of distinct, non-consecutive primes. Squarefree numbers not divisible by any two consecutive primes.
- A325135Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.
- A325134Omega(n) plus the largest prime index of n.
- A325133Heinz number of the integer partition obtained by removing the inner lining, or, equivalently, the largest hook, of the integer partition with Heinz number n.
- A325132Number of integer partitions of n where the multiplicity of each part k is at least prime(k).
- A325131Heinz numbers of integer partitions where the set of distinct parts is disjoint from the set of distinct multiplicities.
- A325130Numbers n in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.
- A325129Heinz numbers of integer partitions into nonsquares (A087153).
- A325128Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.
- A325127Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.
- A325124Number of divisible pairs of positive integers up to n with at least one binary carry.
- A325123Number of divisible pairs of positive integers up to n with no binary carries.
- A325122Sum of binary digits of the prime indices of n, minus Omega(n).
- A325121Sum of binary digits of the prime indices of n.
- A325120Sum of binary lengths of the prime indices of n.
- A325119Heinz numbers of binary carry-connected strict integer partitions.
- A325118Heinz numbers of binary carry-connected integer partitions.
- A325110Number of strict integer partitions of n with no binary containments.
- A325109Number of integer partitions of n whose distinct parts have no binary containments.
- A325108Number of maximal subsets of {1...n} with no binary containments.
- A325107Number of subsets of {1...n} with no binary containments.
- A325106Number of divisible binary-containment pairs of positive integers up to n.
- A325105Number of binary carry-connected subsets of {1...n}.
- A325104Number of increasing pairs of positive integers up to n with at least one binary carry.
- A325103Number of increasing pairs of positive integers up to n with no binary carries.
- A325102Number of ordered pairs of positive integers up to n with no binary carries.
- A325101Number of divisible binary-containment pairs of positive integers up to n.
- A325100Heinz numbers of strict integer partitions with no binary carries.
- A325099Number of binary carry-connected strict integer partitions of n.
- A325098Number of binary carry-connected integer partitions of n.
- A325097Heinz numbers of integer partitions whose distinct parts have no binary carries.
- A325096Number of maximal subsets of {1...n} with no binary carries.
- A325095Number of subsets of {1...n} with no binary carries.
- A325094Write n as a sum of distinct powers of 2, then take the primes of those powers of 2 and multiply them together.
- A325093Heinz numbers of integer partitions into distinct powers of 2.
- A325092Heinz numbers of integer partitions of powers of 2 into powers of 2.
- A325091Heinz numbers of integer partitions of powers of 2.
- A325045Number of factorizations of n whose conjugate as an integer partition has no ones.
- A325044Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.
- A325043Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer than their sum.
- A325042Heinz numbers of integer partitions whose product of parts is one fewer than their sum.
- A325041Heinz numbers of integer partitions whose product of parts is one greater than their sum.
- A325040Heinz numbers of integer partitions with the same product of parts as their conjugate.
- A325039Number of integer partitions of n with the same product of parts as their conjugate.
- A325038Heinz numbers of integer partitions whose sum of parts is greater than their product.
- A325037Heinz numbers of integer partitions whose product of parts is greater than their sum.
- A325036Difference between product and sum of prime indices of n.
- A325035Product of sums of the multisets of prime indices of each prime index of 2 n + 1.
- A325034Sum of products of the multisets of prime indices of each prime index of n.
- A325033Sum of sums of the multisets of prime indices of each prime index of n.
- A325032Product of products of the multisets of prime indices of each prime index of n.
- A325031Numbers divisible by all prime indices of their prime indices.
- A324979Number of rooted trees with n vertices that are not identity trees but whose non-leaf terminal subtrees are all different.
- A324978Matula-Goebel numbers of rooted trees that are not identity trees but whose non-leaf terminal subtrees are all different.
- A324971Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.
- A324970Matula-Goebel numbers of rooted identity trees where not all terminal subtrees are different.
- A324969Number of unlabeled rooted identity trees with n vertices whose non-leaf terminal subtrees are all different.
- A324968Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.
- A324967Number of distinct even prime indices of n.
- A324966Number of distinct odd prime indices of n.
- A324936Number of unlabeled rooted trees with n vertices whose non-leaf terminal subtrees are all different.
- A324935Matula-Goebel numbers of rooted trees whose non-leaf terminal subtrees are all different.
- A324934Inverse permutation of A324931.
- A324933Denominator in the division of n by the product of prime indices of n.
- A324932Numerator in the division of n by the product of prime indices of n.
- A324931Integers in the list of quotients of positive integers by their product of prime indices.
- A324930Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).
- A324929Numbers whose product of prime indices is even.
- A324928Matula-Goebel numbers of rooted trees of depth 3.
- A324927Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.
- A324926Numbers not divisible by any prime indices of their prime indices.
- A324925Number of integer partitions y of n such that Product_{i in y} prime(i)/i is an integer.
- A324924Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.
- A324923Number of distinct factors in the factorization of n into factors q(i) = prime(i)/i, i > 0.
- A324922a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.
- A324856Numbers divisible by exactly one of their prime indices.
- A324855Lexicographically earliest sequence containing 2 and all squarefree numbers > 2 whose prime indices already belong to the sequence.
- A324854Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.
- A324853First number divisible by n distinct prime indices of n.
- A324852Number of distinct prime indices of n that divide n.
- A324851Numbers > 1 divisible by the sum of their prime indices.
- A324850Numbers divisible by the product of their prime indices.
- A324849Positive integers divisible by none of their prime indices > 1.
- A324848Number of prime indices of n (counted with multiplicity) that divide n.
- A324847Numbers divisible by at least one of their prime indices.
- A324846Positive integers divisible by none of their prime indices.
- A324845Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.
- A324844Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.
- A324843Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.
- A324842Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.
- A324841Matula-Goebel numbers of fully recursively anti-transitive rooted trees.
- A324840Number of fully recursively anti-transitive rooted trees with n nodes.
- A324839Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.
- A324838Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.
- A324837Number of minimal subsets of {1...n} with least common multiple n.
- A324771Numbers divisible by at least one of their prime indices > 1.
- A324770Number of fully anti-transitive rooted identity trees with n nodes.
- A324769Matula-Goebel numbers of fully anti-transitive rooted trees.
- A324768Number of fully anti-transitive rooted trees with n nodes.
- A324767Number of recursively anti-transitive rooted identity trees with n nodes.
- A324766Matula-Goebel numbers of recursively anti-transitive rooted trees.
- A324765Number of recursively anti-transitive rooted trees with n nodes.
- A324764Number of anti-transitive rooted identity trees with n nodes.
- A324763Number of maximal subsets of {2...n} containing no prime indices of the elements.
- A324762Number of maximal subsets of {2...n} containing no element whose prime indices all belong to the subset.
- A324761Heinz numbers of integer partitions not containing 1 or any prime indices of the parts.
- A324760Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.
- A324759Heinz numbers of integer partitions containing no part > 1 whose prime indices all belong to the partition.
- A324758Heinz numbers of integer partitions containing no prime indices of the parts.
- A324757Number of integer partitions of n not containing 1 or any prime indices of the parts.
- A324756Number of integer partitions of n containing no prime indices of the parts.
- A324755Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.
- A324754Number of integer partitions of n containing no part > 1 whose prime indices all belong to the partition.
- A324753Number of integer partitions of n containing all prime indices of their parts.
- A324752Number of strict integer partitions of n not containing 1 or any prime indices of the parts.
- A324751Number of strict integer partitions of n containing no prime indices of the parts.
- A324750Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.
- A324749Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.
- A324748Number of strict integer partitions of n containing all prime indices of the parts.
- A324744Number of maximal subsets of {1...n} containing no element whose prime indices all belong to the subset.
- A324743Number of maximal subsets of {1...n} containing no prime indices of the elements.
- A324742Number of subsets of {2...n} containing no prime indices of the elements.
- A324741Number of subsets of {1...n} containing no prime indices of the elements.
- A324739Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.
- A324738Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.
- A324737Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.
- A324736Number of subsets of {1...n} containing all prime indices of the elements.
- A324705Lexicographically earliest sequence containing 1 and all composite numbers divisible by prime(m) for some m already in the sequence.
- A324704Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.
- A324703Lexicographically earliest sequence containing 3 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.
- A324702Lexicographically earliest sequence containing 2 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.
- A324701Lexicographically earliest sequence containing 1 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.
- A324700Lexicographically earliest sequence containing 0 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.
- A324699Lexicographically earliest sequence of positive integers whose prime indices minus 1 already belong to the sequence.
- A324698Lexicographically earliest sequence containing 2 and all numbers > 1 whose prime indices already belong to the sequence.
- A324697Lexicographically earliest sequence of positive integers > 1 that are prime or whose prime indices already belong to the sequence.
- A324696Lexicographically earliest sequence containing 1 and all numbers divisible by prime(m) for some m not already in the sequence.
- A324695Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.
- A324694Lexicographically earliest sequence of positive integers divisible by prime(m) for some m not already in the sequence.
- A324588Heinz numbers of integer partitions of n into perfect squares (A001156).
- A324587Heinz numbers of integer partitions of n into distinct perfect squares (A033461).
- A324572Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.
- A324571Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.
- A324570Numbers where the sum of distinct prime indices (A066328) is equal to the number of prime factors counted with multiplicity (A001222).
- A324562Numbers > 1 where the maximum prime index is greater than or equal to the number of prime factors counted with multiplicity.
- A324561Numbers with at least one prime index equal to 0, 1, or 4 modulo 5.
- A324560Numbers > 1 where the minimum prime index is less than or equal to the number of prime factors counted with multiplicity.
- A324525Numbers in whose prime factorization the multiplicity of prime(k) is at least k. Numbers divisible by prime(k)^k for each prime index k.
- A324524Numbers in whose prime factorization the exponent of prime(k) is a multiple of k. Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.
- A324522Numbers > 1 where the minimum prime index is equal to the number of prime factors counted with multiplicity.
- A324521Numbers > 1 where the maximum prime index is less than or equal to the number of prime factors counted with multiplicity.
- A324520Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.
- A324519Numbers > 1 where the minimum prime index equals the number of prime factors minus the number of distinct prime factors.
- A324518Number of integer partitions of n > 0 where the maximum part equals the length minus the number of distinct parts.
- A324517Numbers > 1 where the maximum prime index equals the number of prime factors minus the number of distinct prime factors.
- A324516Number of integer partitions of n > 0 where the maximum part minus the minimum part equals the length minus the number of distinct parts.
- A324515Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.
- A324514Number of aperiodic permutations of {1...n}.
- A324513Number of aperiodic cycle necklaces with n vertices.
- A324512Number of aperiodic n-gons.
- A324464Number of connected graphical necklaces with n vertices.
- A324463Number of graphical necklaces covering n vertices.
- A324462Number of simple graphs covering n vertices with distinct rotations.
- A324461Number of simple graphs with n vertices and distinct rotations.
- A324328Number of topologically connected chord graphs on a subset of {1,...,n}.
- A324327Number of topologically connected chord graphs covering {1,...,n}.
- A324326Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A324325Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A324324MM-numbers of crossing set partitions.
- A324323Regular triangle read by rows where T(n,k) is the number of topologically connected set partitions of {1,...,n} with k blocks, 0 <= k <= n.
- A324173Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.
- A324171Number of non-crossing multiset partitions of normal multisets of size n.
- A324170Numbers whose multiset multisystem (A302242) is crossing.
- A324169Number of labeled graphs covering the vertex set {1,...,n} with no crossing edges.
- A324168Number of non-crossing antichains of nonempty subsets of {1,...,n}.
- A324167Number of non-crossing antichain covers of {1,...,n}.
- A324166Number of totally crossing set partitions of {1,...,n}.
- A324015Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.
- A324014Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).
- A324013Number of self-complementary set partitions of {1, ..., n} with no singletons.
- A324012Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).
- A324011Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).
- A323956Regular triangle read by rows where T(n, k) = 1 + n * (n - k + 1), n >=1, 2 <= k <= n + 1.
- A323955Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.
- A323954Regular triangle read by rows where T(n, k) is the number of of ways to split an n-cycle into connected subsequences of sizes > k, n >=1, 0 <= k < n.
- A323953Regular triangle read by rows where T(n, k) is the number of ways to split an n-cycle into singletons and connected subsequences of sizes > k.
- A323952Regular triangle read by rows where if k > 1 then T(n, k) is the number of connected subgraphs of an n-cycle with any number of vertices other than 2 through k - 1, n >= 1, 1 <= k <= n - 1. Otherwise, T(n, 1) = n.
- A323951Number of ways to split an n-cycle into connected subgraphs, all having at least three vertices.
- A323950Number of ways to split an n-cycle into connected subgraphs, none having exactly two vertices.
- A323949Number of set partitions of {1, ..., n} with no block containing three distinct cyclically successive vertices.
- A323872Number of n X n aperiodic binary toroidal necklaces.
- A323871Number of aperiodic toroidal necklaces of size n whose entries cover an initial interval of positive integers.
- A323870Number of toroidal necklaces of size n whose entries cover an initial interval of positive integers.
- A323869Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.
- A323868Number of matrices of size n whose entries cover an initial interval of positive integers.
- A323867Number of aperiodic arrays of positive integers summing to n.
- A323866Number of aperiodic toroidal necklaces of positive integers summing to n.
- A323865Number of aperiodic binary toroidal necklaces of size n.
- A323864Number of aperiodic binary arrays of size n.
- A323863Number of n X n aperiodic binary arrays.
- A323862Table read by antidiagonals where A(n,k) is the number of n X k binary arrays in which both the sequence of rows and the sequence of columns are (independently) aperiodic.
- A323861Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.
- A323860Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.
- A323859Number of binary toroidal necklaces of size n.
- A323858Number of toroidal necklaces of positive integers summing to n.
- A323820Number of non-isomorphic connected set-systems covering n vertices with no singletons.
- A323819Number of non-isomorphic connected set-systems covering n vertices.
- A323818Number of connected set-systems covering n vertices.
- A323817Number of connected set-systems covering n vertices with no singletons.
- A323816Number of set-systems covering n vertices with no singletons.
- A323795Number of non-isomorphic weight-n sets of non-overlapping sets of sets.
- A323794Number of non-isomorphic weight-n multisets of sets of multisets.
- A323793Number of non-isomorphic weight-n multisets of multisets of sets.
- A323792Number of non-isomorphic weight-n multisets of sets of sets.
- A323791Number of non-isomorphic weight-n sets of multisets of sets.
- A323790Number of non-isomorphic weight-n sets of sets of sets.
- A323789Number of non-isomorphic weight-n sets of sets of multisets.
- A323788Number of non-isomorphic weight-n sets of multisets of multisets.
- A323787Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.
- A323786Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.
- A323776a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).
- A323775a(n) = Sum_{k = 1...n} k^(2^(n - k)).
- A323774Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.
- A323766Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.
- A323765Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.
- A323764Dirichlet self-convolution of the integer partition numbers A000041.
- A323719Array read by antidiagonals where T(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.
- A323718Array read by antidiagonals where T(n, k) is the number of k-times partitions of n.
- A323657Number of strict solid partitions of n.
- A323656Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.
- A323655Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.
- A323654Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.
- A323587Number of strict (distinct parts) plane partitions of n with relatively prime parts.
- A323586Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).
- A323585Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.
- A323584Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.
- A323583Number of ways to split an integer partition of n into consecutive subsequences.
- A323582Number of generalized Young tableaux with constant rows, weakly increasing columns, and entries summing to n.
- A323581Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.
- A323580Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.
- A323531Number of square multiset partitions of integer partitions of n.
- A323530Number of square plane partitions of n with strictly decreasing rows and columns.
- A323529Number of strict square plane partitions of n.
- A323528Numbers whose sum of prime indices is a perfect square.
- A323527Numbers whose sum of prime indices is not a perfect square.
- A323526One and prime numbers indexed by perfect squares.
- A323525Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.
- A323524Number of integer partitions of n whose parts can be arranged into a square matrix with equal row and column sums.
- A323523Number of positive integer square matrices with entries summing to n and equal row and column sums.
- A323522Number of ways to fill a square matrix with the parts of a strict integer partition of n.
- A323521Numbers whose number of prime factors counted with multiplicity (A001222) is not a perfect square.
- A323520Numbers of the form p^(k^2) where p is prime and k >= 0.
- A323519Number of ways to fill a square matrix with the multiset of prime factors of n.
- A323451Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.
- A323450Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.
- A323440Numbers divisible by exactly one of their distinct prime indices.
- A323439Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.
- A323438Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.
- A323437Number of semistandard Young tableaux whose entries are the prime indices of n.
- A323436Number of plane partitions whose parts are the prime indices of n.
- A323435Number of rectangular plane partitions of n with no repeated rows or columns.
- A323434Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.
- A323433Number of ways to split an integer partition of n into consecutive subsequences of equal length.
- A323432Number of semistandard rectangular plane partitions of n.
- A323431Number of strict rectangular plane partitions of n.
- A323430Number of rectangular plane partitions of n with strictly decreasing rows and columns.
- A323429Number of rectangular plane partitions of n.
- A323351Number of ways to fill a (not necessarily square) matrix with n zeros and ones.
- A323350Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.
- A323349Number of positive integer matrices with entries summing to n with equal row-sums and equal column-sums.
- A323348Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.
- A323347Number of integer partitions of n whose parts can be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.
- A323307Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.
- A323306Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.
- A323305Number of divisors of the number of prime factors of n counted with multiplicity.
- A323304Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.
- A323303Number of ways to arrange the prime indices of n into a matrix with equal column-sums.
- A323302Number of ways to arrange the parts of the integer partition with Heinz number n into a matrix with equal row-sums and equal column-sums.
- A323301Number of ways to fill a matrix with the parts of a strict integer partition of n.
- A323300Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.
- A323299Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.
- A323298Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.
- A323297Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.
- A323296Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.
- A323295Number of ways to fill a matrix with the first n positive integers.
- A323294Number of 3-uniform hypergraphs spanning n labeled vertices in which every two edges have two vertices in common.
- A323293Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.
- A323292Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.
- A323094Number of strict integer partitions of n where no part is 2^k times any other part, for any k > 0.
- A323093Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.
- A323092Number of double-free integer partitions of n.
- A323091Number of strict knapsack factorizations of n.
- A323090Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).
- A323089Number of strict integer partitions of n using 1 and numbers that are not perfect powers.
- A323088Number of strict integer partitions of n using numbers that are not perfect powers.
- A323087Number of strict factorizations of n into factors > 1 such that no factor is a power of any other factor.
- A323086Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.
- A323056Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.
- A323055Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.
- A323054Number of strict integer partitions of n with no 1's such that no part is a power of any other part.
- A323053Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.
- A323025Numbers with exactly four distinct exponents in their prime factorization, or four distinct parts in their prime signature.
- A323024Numbers with exactly three distinct exponents in their prime factorization, or three distinct parts in their prime signature.
- A323023Irregular triangle read by rows where row n is the omega-sequence of n.
- A323022Fourth omega of n. Number of distinct multiplicities in the prime signature of n.
- A323014a(1) = 0; a(prime) = 1; otherwise a(n) = 1 + a(A181819(n)).
- A322968Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.
- A322912Number of integer partitions of n whose parts are all powers of the same squarefree number.
- A322911Numbers whose prime indices are all powers of the same squarefree number.
- A322903Odd numbers whose prime indices are all proper powers of the same number.
- A322902Numbers whose prime indices are all proper powers of the same number.
- A322901Numbers whose prime indices are all powers of the same number.
- A322900Number of integer partitions of n whose parts are all proper powers of the same number.
- A322847Numbers whose prime indices have no equivalent primes.
- A322846Squarefree numbers whose prime indices have no equivalent primes.
- A322841Number of positive integers less than n with more distinct prime factors than n.
- A322840Positive integers n with fewer prime factors than n + 1, counted with multiplicity.
- A322839Numbers n with more prime factors than n+1, counted with multiplicity.
- A322838Number of positive integers less than n with more prime factors than n, counted with multiplicity.
- A322837Number of positive integers less than n with fewer distinct prime factors than n.
- A322833Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.
- A322794Number of factorizations of n into factors > 1 where all factors have the same number of prime factors counted with multiplicity.
- A322793Proper powers of primorial numbers.
- A322792Irregular triangle read by rows where if d|n then T(n,d) = A002110(n/d)^d, where A002110(k) is the product of the first k primes.
- A322789Irregular triangle read by rows where if d|n then T(n,d) is the number of non-isomorphic uniform multiset partitions of a multiset with d copies of each integer from 1 to n/d.
- A322788Irregular triangle read by rows where if d|n then T(n,d) is the number of uniform multiset partitions of a multiset with d copies of each integer from 1 to n/d.
- A322787Irregular triangle read by rows where if d|n then T(n,d) is the number of non-isomorphic multiset partitions of a multiset with d copies of each integer from 1 to n/d.
- A322786Irregular triangle read by rows where if d|n then T(n,d) is the number of multiset partitions of a multiset with d copies of each integer from 1 to n/d.
- A322785Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.
- A322784Number of multiset partitions of uniform multisets of weight n whose union is an initial interval of positive integers.
- A322706Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.
- A322705Number of k-uniform k-regular hypergraphs spanning n labeled vertices, for some 1 <= k <= n.
- A322704Number of regular hypergraphs on n labeled vertices with no singletons.
- A322703Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.
- A322700Number of unlabeled graphs with loops spanning n vertices.
- A322698Number of regular graphs with half-edges on n labeled vertices.
- A322661Number of graphs with loops spanning n labeled vertices.
- A322659Number of connected regular simple graphs on n labeled vertices.
- A322635Number of regular graphs with loops on n labeled vertices.
- A322555Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.
- A322554Numbers whose product of prime indices is a power of a squarefree number (A072774).
- A322553Odd numbers whose product of prime indices is a prime power.
- A322552MM-numbers of triangles.
- A322551Primes indexed by squarefree semiprimes.
- A322547Numbers n such that every integer partition of n contains a 1, a squarefree number, or a prime power.
- A322546Numbers n such that every integer partition of n contains a 1 or a prime power.
- A322531Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.
- A322530Number of integer partitions of n with no 1's whose product of parts is a squarefree number.
- A322529Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.
- A322528Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).
- A322527Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).
- A322526Number of integer partitions of n whose product of parts is a squarefree number.
- A322454Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.
- A322453Number of factorizations of n into factors > 1 using only primes and perfect powers.
- A322452Number of factorizations of n into factors > 1 not including any prime powers.
- A322451Number of unlabeled 3-regular hypergraphs spanning n vertices.
- A322442Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.
- A322441Number of pairs of set partitions of {1,...,n} where no block of one is a subset or equal to any block of the other.
- A322440Number of pairs of integer partitions of n where every part of the first is less than every part of the second.
- A322439Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.
- A322438Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.
- A322437Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.
- A322436Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.
- A322435Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.
- A322401Number of strict integer partitions of n with edge-connectivity 1.
- A322400Heinz numbers of integer partitions with vertex-connectivity 1.
- A322399Number of non-isomorphic 2-edge-connected clutters spanning n vertices.
- A322397Number of 2-edge-connected clutters spanning n vertices.
- A322396Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.
- A322395Number of labeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.
- A322394Heinz numbers of integer partitions with edge-connectivity 1.
- A322393Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n.
- A322391Number of integer partitions of n with edge-connectivity 1.
- A322390Number of integer partitions of n with vertex-connectivity 1.
- A322389Vertex-connectivity of the integer partition with Heinz number n.
- A322388Heinz numbers of 2-vertex-connected integer partitions.
- A322387Number of 2-vertex-connected integer partitions of n.
- A322386Numbers whose prime indices are not prime and already belong to the sequence.
- A322385Prime numbers whose prime index is a nonprime product of prime numbers already in the sequence.
- A322369Number of strict disconnected or empty integer partitions of n.
- A322368Heinz numbers of disconnected integer partitions.
- A322367Number of disconnected or empty integer partitions of n.
- A322338Edge-connectivity of the integer partition with Heinz number n.
- A322337Number of strict 2-edge-connected integer partitions of n.
- A322336Heinz numbers of 2-edge-connected integer partitions.
- A322335Number of 2-edge-connected integer partitions of n.
- A322307Number of multisets in the swell of the n-th multiset multisystem.
- A322306Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).
- A322260Numbers n such that the poset of multiset partitions of a multiset whose multiplicities are the prime indices of n is a lattice.
- A322151Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).
- A322148Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.
- A322147Regular triangle where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.
- A322141Number of unlabeled 2-connected multigraphs with n edges.
- A322140Number of labeled 2-connected multigraphs with n edges (the vertices are {1,2,...,k} for some k).
- A322139Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).
- A322138Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.
- A322137Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).
- A322136Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.
- A322134Regular tetrangle where T(n,k,i) is the number of unlabeled connected multiset partitions of weight n with k vertices and i edges.
- A322133Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.
- A322118Number of non-isomorphic 2-connected multiset partitions of weight n with no singletons.
- A322117Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets.
- A322115Regular triangle where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.
- A322114Regular triangle where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.
- A322113Number of non-isomorphic self-dual connected antichains of multisets of weight n.
- A322112Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.
- A322111Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.
- A322110Number of non-isomorphic 2-connected multiset partitions of weight n.
- A322109Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.
- A322077In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition whose multiplicities are the prime indices of n in weakly decreasing order.
- A322076Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.
- A322075Number of factorizations of n into nonprime squarefree numbers > 1.
- A322066Number of e-positive antichains of sets spanning n vertices.
- A322065Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.
- A322064Number of ways to choose a stable partition of a simple connected graph with n vertices.
- A322063Number of ways to choose a stable partition of an antichain of sets spanning n vertices.
- A322030Numbers whose prime factors all have the same order of primeness.
- A322028Number of distinct orders of primeness among the prime factors of n.
- A322027Maximum order of primeness among the prime factors of n.
- A322014Heinz numbers of integer partitions with an even number of even parts.
- A322012Number of s-positive simple labeled graphs with n vertices.
- A322011Number of distinct chromatic symmetric functions of spanning hypergraphs (or antichain covers) on n vertices.
- A321994Number of different chromatic symmetric functions of hypertrees on n vertices.
- A321982Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.
- A321981Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.
- A321980Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.
- A321979Number of e-positive simple labeled graphs on n vertices.
- A321936Number of integer partitions of n containing no 1's, prime powers, or squarefree numbers.
- A321935Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and S is augmented Schur functions.
- A321934Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.
- A321933Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.
- A321932Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in e(u) * Product_i u_i!, where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and e is elementary symmetric functions.
- A321931Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.
- A321930Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in f(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.
- A321929Tetrangle where T(n,H(u),H(v)) is the coefficient of f(v) in s(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.
- A321928Tetrangle where T(n,H(u),H(v)) is the coefficient of f(v) in p(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and p is power sum symmetric functions.
- A321927Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in f(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and f is forgotten symmetric functions.
- A321926Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in p(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and p is power sum symmetric functions.
- A321925Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in m(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and m is monomial symmetric functions.
- A321924Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in s(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and s is Schur functions.
- A321923Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in h(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and h is homogeneous symmetric functions.
- A321922Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in s(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.
- A321921Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in e(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and e is elementary symmetric functions.
- A321920Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in s(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and s is Schur functions.
- A321919Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.
- A321918Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.
- A321917Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in p(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.
- A321916Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.
- A321915Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.
- A321914Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.
- A321913Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.
- A321912Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in e(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.
- A321911Number of distinct chromatic symmetric functions of simple connected graphs with n vertices.
- A321908If y is the integer partition with Heinz number n, then a(n) = |y|! / syt(y), where syt(y) is the number of standard Young tableaux of shape y.
- A321907If n > 1 is the k-th prime number, then a(n) = k!, otherwise a(n) = 0.
- A321900Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in S(u), where H is Heinz number, p is power sum symmetric functions, and S is augmented Schur functions.
- A321899Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in F(u), where H is Heinz number, F is augmented forgotten symmetric functions, and p is power sum symmetric functions.
- A321898Sum of coefficients of power sums symmetric functions in h(y) * Product_i y_i! where h is homogeneous symmetric functions and y is the integer partition with Heinz number n.
- A321897Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.
- A321896Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in e(u) * Product_i u_i!, where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.
- A321895Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in M(u), where H is Heinz number, M is augmented monomial symmetric functions, and p is power sum symmetric functions.
- A321894Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in f(u), where H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.
- A321893Sum of coefficients of forgotten symmetric functions in the Schur function of the integer partition with Heinz number n.
- A321892Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in s(u), where H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.
- A321889Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.
- A321888Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and f is forgotten symmetric functions.
- A321887Sum of coefficients of monomial symmetric functions in the forgotten symmetric function indexed by the integer partition with Heinz number n.
- A321886Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in f(u), where H is Heinz number, m is monomial symmetric functions, and f is forgotten symmetric functions.
- A321854Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.
- A321765Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and s is Schur functions.
- A321764Sum of coefficients of Schur functions in the monomial symmetric function indexed by the integer partition with Heinz number n.
- A321763Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and s is Schur functions.
- A321762Sum of coefficients of monomial symmetric functions in the Schur function of the integer partition with Heinz number n.
- A321761Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in s(u), where H is Heinz number, m is monomial symmetric functions, and s is Schur functions.
- A321760Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.
- A321759Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in h(u), where H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.
- A321758Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in s(u), where H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.
- A321757Sum of coefficients of Schur functions in the elementary symmetric function of the integer partition with Heinz number n.
- A321756Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in e(u), where H is Heinz number, e is elementary symmetric functions, and s is Schur functions.
- A321755Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in s(u), where H is Heinz number, e is elementary symmetric functions, and s is Schur functions.
- A321754Irregular triangle where T(H(u),H(v)) is the coefficient of h(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.
- A321753Sum of coefficients of elementary symmetric functions in the power sum symmetric function indexed by the integer partition with Heinz number n.
- A321752Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in p(u), where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.
- A321751Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.
- A321750Irregular triangle where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.
- A321749Irregular triangle where T(H(u),H(v)) is the coefficient of e(v) in h(u) or, equivalently, the coefficient of h(v) in e(u), where H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.
- A321748Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.
- A321747Sum of coefficients of elementary symmetric functions in the monomial symmetric function of the integer partition with Heinz number n.
- A321746Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.
- A321745Sum of coefficients of monomial symmetric functions in the homogeneous symmetric function of the integer partition with Heinz number n.
- A321744Irregular triangle where T(H(u),H(v)) is the coefficient of m(v) in h(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.
- A321743Sum of coefficients of monomial symmetric functions in the elementary symmetric function of the integer partition with Heinz number n.
- A321742Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in e(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.
- A321739Number of non-isomorphic weight-n set multipartitions (multisets of sets) whose part-sizes are also their vertex-degrees.
- A321738Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.
- A321737Number of ways to partition the Young diagram of an integer partition of n into vertical sections.
- A321736Number of non-isomorphic weight-n multiset partitions whose part-sizes are also their vertex-degrees.
- A321735Number of (0,1)-matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.
- A321734Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.
- A321733Number of (0,1)-matrices with n ones, no zero rows or columns, and the same row sums as column sums.
- A321732Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, and the same row sums as column sums.
- A321731Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections of the same sizes as the parts of the original partition.
- A321730Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.
- A321729Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.
- A321728Number of integer partitions of n whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition.
- A321725Irregular triangle read by rows where T(n,d) is the number of d X d non-normal semi-magic squares with sum of all entries equal to n.
- A321724Irregular triangle read by rows where T(n,d) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d|n.
- A321723Number of non-normal magic squares whose entries are all 0 or 1 and sum to n.
- A321722Number of non-normal magic squares whose entries are nonnegative integers summing to n.
- A321721Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.
- A321720Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.
- A321719Number of non-normal semi-magic squares with sum of entries equal to n.
- A321718Number of coupled non-normal semi-magic rectangles with sum of entries equal to n.
- A321717Number of non-normal (0,1) semi-magic rectangles with sum of all entries equal to n.
- A321699MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.
- A321698MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.
- A321681Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.
- A321680Number of non-isomorphic weight-n connected antichains (not necessarily strict) of multisets with multiset density -1.
- A321679Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.
- A321678Number of non-isomorphic weight-n strict antichains of sets with no singletons.
- A321677Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.
- A321665Number of strict integer partitions of n containing no 1's or prime powers.
- A321662Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.
- A321661Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.
- A321660Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.
- A321659Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose nonzero entries are all distinct.
- A321655Number of distinct row/column permutations of strict plane partitions of n.
- A321654Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with distinct row sums and distinct column sums.
- A321653Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with strictly decreasing row sums and column sums.
- A321652Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.
- A321650Irregular triangle whose n-th row is the reversed conjugate of the integer partition with Heinz number n.
- A321649Irregular triangle whose n-th row is the conjugate of the integer partition with Heinz number n.
- A321648Number of permutations of the conjugate of the integer partition with Heinz number n.
- A321647Number of distinct row/column permutations of the Ferrers diagram of the integer partition with Heinz number n.
- A321646Number of distinct row/column permutations of Ferrers diagrams of integer partitions of n.
- A321645Number of distinct row/column permutations of plane partitions of n.
- A321588Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.
- A321587Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows.
- A321586Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows (or distinct columns).
- A321585Number of connected nonnegative integer matrices with sum of entries equal to n and no zero rows or columns.
- A321584Number of connected (0,1)-matrices with n ones and no zero rows or columns.
- A321515Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.
- A321514Number of ways to choose a factorization of each integer from 2 to n into factors > 1.
- A321484Number of non-isomorphic self-dual connected multiset partitions of weight n.
- A321472Heinz numbers of integer partitions whose parts can be further partitioned and flattened to obtain the partition (k, ..., 3, 2, 1) for some k.
- A321471Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.
- A321470Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining.
- A321469Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.
- A321468Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.
- A321467Number of factorizations of n! into factors > 1 that can be obtained by taking the block-products of some set partition of {2,...,n}.
- A321455Number of ways to factor n into factors > 1 all having the same sum of prime indices.
- A321454Numbers that can be factored into two or more factors all having the same sum of prime indices.
- A321453Numbers that cannot be factored into two or more factors all having the same sum of prime indices.
- A321452Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.
- A321451Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.
- A321449Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts.
- A321446Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows and columns.
- A321413Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.
- A321412Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.
- A321411Number of non-isomorphic self-dual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.
- A321410Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets whose sizes are relatively prime.
- A321409Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.
- A321408Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic.
- A321407Number of non-isomorphic multiset partitions of weight n with no constant parts.
- A321406Number of non-isomorphic self-dual set systems of weight n with no singletons.
- A321405Number of non-isomorphic self-dual set systems of weight n.
- A321404Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n with no singletons.
- A321403Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n.
- A321402Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.
- A321390Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.
- A321378Number of integer partitions of n containing no 1's or prime powers.
- A321347Number of strict integer partitions of n containing no prime powers (including 1).
- A321346Number of integer partitions of n containing no prime powers > 1.
- A321283Number of non-isomorphic multiset partitions of weight n in which the part sizes are relatively prime.
- A321279Number of z-trees with product A181821(n). Number of connected antichains of multisets with multiset density -1, of a multiset whose multiplicities are the prime indices of n.
- A321272Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.
- A321271Number of connected factorizations of n into positive integers > 1 with z-density -1.
- A321270Number of connected multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A321256Regular triangle where T(n,k) is the number of non-isomorphic connected set systems of weight n with density -1 <= k <= n-2.
- A321255Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n, with no singletons.
- A321254Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.
- A321253Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.
- A321231Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.
- A321229Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.
- A321228Number of non-isomorphic hypertrees of weight n with singletons.
- A321227Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.
- A321194Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.
- A321188Number of set systems with no singletons whose multiset union is a row n of A305936 (a multiset whose multiplicities are the prime indices of n).
- A321185Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.
- A321184Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.
- A321177Heinz numbers of integer partitions that are the vertex-degrees of some set system with no singletons.
- A321176Number of integer partitions of n that are the vertex-degrees of some set system with no singletons.
- A321155Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with density -1 <= k < n-2.
- A321144Irregular triangle where T(n,k) is the number of divisors of n whose prime indices sum to k.
- A321143Number of non-isomorphic knapsack multiset partitions of weight n.
- A321142Number of strict integer partitions of 2*n with no subset summing to n.
- A321134Number of uniform hypergraphs spanning n vertices where every two vertices appear together in some edge.
- A320925Heinz numbers of connected multigraphical partitions.
- A320924Heinz numbers of multigraphical partitions.
- A320923Heinz numbers of connected graphical partitions.
- A320922Heinz numbers of graphical partitions.
- A320921Number of connected graphical partitions of 2n.
- A320913Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes (A320891) but can be factored into distinct semiprimes (A320912).
- A320912Numbers with an even number of prime factors (counted with multiplicity) that can be factored into distinct semiprimes.
- A320911Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.
- A320894Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct squarefree semiprimes.
- A320893Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes (A320911) but cannot be factored into distinct semiprimes (A320892).
- A320892Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.
- A320891Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.
- A320889Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.
- A320888Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.
- A320887Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.
- A320886Number of multiset partitions of integer partitions of n where all parts have the same product.
- A320836a(n) = Sum (-1)^k where the sum is over all strict multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or strict factorizations of A181821(n).
- A320835a(n) = Sum (-1)^k where the sum is over all multiset partitions of a multiset whose multiplicities are the prime indices of n and k is the number of parts, or factorizations of A181821(n).
- A320813Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.
- A320812Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.
- A320811Number of non-isomorphic multiset partitions with no singletons of aperiodic multisets of size n.
- A320810Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.
- A320809Number of non-isomorphic multiset partitions of weight n in which each part and each part of the dual, as well as the multiset union of the parts, is an aperiodic multiset.
- A320808Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n and i columns.
- A320807Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic.
- A320806Number of non-isomorphic multiset partitions of weight n in which each of the parts and each part of the dual, as well as both the multiset union of the parts and the multiset union of the dual parts, is an aperiodic multiset.
- A320805Number of non-isomorphic multiset partitions of weight n in which each part, as well as the multiset union of the parts, is an aperiodic multiset.
- A320804Number of non-isomorphic multiset partitions of weight n with no singletons in which all parts are aperiodic multisets.
- A320803Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic multisets.
- A320802Number of non-isomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition.
- A320801Regular triangle where T(n,k) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n.
- A320800Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.
- A320799Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).
- A320798Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.
- A320797Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.
- A320796Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.
- A320786Inverse Euler transform of {1,0,1,0,0,0,...}
- A320785Inverse Euler transform of the number of factorizations function A001055.
- A320784Negated inverse Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054.
- A320783Inverse Euler transform of (-1)^(n - 1).
- A320782Inverse Euler transform of the unsigned Moebius function A008966.
- A320781Inverse Euler transform of the Moebius function A008683.
- A320780Inverse Euler transform of the sum-of-divisors or sigma function A000203.
- A320779Inverse Euler transform of the number of divisors function A000005.
- A320778Inverse Euler transform of the Euler totient function phi = A000010.
- A320777Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.
- A320776Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.
- A320768Number of set partitions of the set of nonempty subsets of {1,...,n} using set partitions.
- A320767Inverse Euler transform applied once to {1,-1,0,0,0,...}, twice to {1,0,0,0,0,...}, or three times to {1,1,1,1,1,...}.
- A320732Number of factorizations of n into primes or semiprimes.
- A320700Odd numbers whose product of prime indices is a non-prime prime power (A246547).
- A320699Numbers whose product of prime indices is a non-prime prime power (A246547).
- A320698Numbers whose product of prime indices is a prime power (A246655).
- A320665Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.
- A320664Number of non-isomorphic multiset partitions of weight n with all parts of odd size.
- A320663Number of non-isomorphic multiset partitions of weight n using singletons or pairs.
- A320659Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.
- A320658Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs.
- A320656Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.
- A320655Number of factorizations of n into semiprimes. Number of multiset partitions of the multiset of prime factors of n, into pairs.
- A320635MM-numbers of labeled connected simple graphs spanning an initial interval of positive integers.
- A320634Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).
- A320633Composite numbers whose prime indices are also composite.
- A320632Numbers n such that there exists a pair of factorizations of n into factors > 1 where no factor of one divides any factor of the other.
- A320631Products of odd primes of nonprime squarefree index.
- A320630Products of primes of nonprime squarefree index.
- A320629Products of odd primes of nonprime index.
- A320628Products of primes of nonprime index.
- A320606Regular triangle read by rows where T(n,k) is the number of k-uniform hypergraphs spanning n labeled vertices where every two vertices appear together in some edge, n >= 0, 0 <= k <= n.
- A320533MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.
- A320532MM-numbers of labeled hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.
- A320464MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.
- A320463MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.
- A320462MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.
- A320461MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.
- A320459MM-numbers of labeled multigraphs spanning an initial interval of positive integers.
- A320458MM-numbers of labeled simple graphs spanning an initial interval of positive integers.
- A320456Numbers whose multiset multisystem spans an initial interval of positive integers.
- A320451Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.
- A320450Number of strict antichains of sets whose multiset union is an integer partition of n.
- A320449Number of antichains of sets whose multiset union is an integer partition of n.
- A320446Covers of triangles by tetrahedra: number of labeled 4-uniform hypergraphs spanning n vertices such that every three vertices appear together in some edge.
- A320444Number of uniform hypertrees spanning n vertices.
- A320439Number of factorizations of n into factors > 1 where each factor's prime indices are relatively prime. Number of factorizations of n using elements of A289509.
- A320438Irregular triangle read by rows where T(n,d) is the number of set partitions of {1,...n} with all block-sums equal to d, where d is a divisor of 1 + ... + n.
- A320436Irregular triangle read by rows where T(n,k) is the number of pairwise coprime k-subsets of {1,...,n}, 1 <= k <= A036234(n), where a single number is not considered to be pairwise coprime unless it is equal to 1.
- A320435Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.
- A320430Number of set partitions of {1,...,n} where the elements of each non-singleton block are pairwise coprime.
- A320426Number of nonempty pairwise coprime subsets of {1,...,n}, where a single number is not considered to be pairwise coprime unless it is equal to 1.
- A320424Number of set partitions of {1,...,n} where each block's elements are relatively prime.
- A320423Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.
- A320395Number of non-isomorphic 3-uniform multiset systems over {1,...,n}.
- A320356Number of strict connected antichains of multisets whose multiset union is an integer partition of n.
- A320355Number of connected antichains of multisets whose multiset union is an integer partition of n.
- A320353Number of antichains of multisets whose multiset union is an integer partition of n.
- A320351Number of connected multiset partitions of integer partitions of n.
- A320340Heinz numbers of double-free integer partitions.
- A320331Number of strict T_0 multiset partitions of integer partitions of n.
- A320330Number of T_0 multiset partitions of integer partitions of n.
- A320328Number of square multiset partitions of integer partitions of n.
- A320325Numbers whose product of prime indices is a perfect power.
- A320324Numbers of which each prime index has the same number of prime factors, counted with multiplicity.
- A320323Numbers whose product of prime indices (A003963) is a perfect power and where each prime index has the same number of prime factors, counted with multiplicity.
- A320322Number of integer partitions of n whose product [of parts] is a perfect power.
- A320296Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.
- A320295Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.
- A320294Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n with no 1's.
- A320293Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n with no 1's.
- A320291Number of singleton-free multiset partitions of integer partitions of n with no 1's.
- A320289Number of phylogenetic trees with n labels and no singleton leaves.
- A320275Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.
- A320271Number of unlabeled semi-binary rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.
- A320270Number of unlabeled balanced semi-binary rooted trees with n nodes.
- A320269Matula-Goebel numbers of series-reduced rooted trees in which the non-leaf branches directly under any given node are all equal.
- A320268Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.
- A320267Number of balanced complete orderless tree-factorizations of n.
- A320266Number of balanced orderless tree-factorizations of n.
- A320230Matula-Goebel numbers of rooted trees in which the non-leaf branches directly under any given node are all equal.
- A320226Number of integer partitions of n whose non-1 parts are all equal.
- A320225a(1) = 1; a(n > 1) = Sum_{k = 1...n} Sum_{d|k, d < k} a(d).
- A320224a(1) = 1; a(n > 1) = Sum_{k = 1...n-1} Sum_{d|k, d < k} a(d).
- A320222Number of unlabeled rooted trees with n nodes in which the non-leaf branches directly under any given node are all equal.
- A320221Irregular triangle with zeros removed where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.
- A320179Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k, where 0 <= k <= n-1.
- A320178Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.
- A320177Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.
- A320176Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.
- A320175Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.
- A320174Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.
- A320173Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.
- A320172Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
- A320171Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
- A320169Number of balanced enriched p-trees of weight n.
- A320167Regular triangle where T(n,k) = Sum (-1)^i where the sum is over all factorizations of n into i factors that are all <= k.
- A320160Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.
- A320155Number of series-reduced balanced rooted trees with n labeled leaves.
- A320154Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.
- A320058Heinz numbers of spanning product-sum knapsack partitions.
- A320057Heinz numbers of spanning sum-product knapsack partitions.
- A320056Heinz numbers of product-sum knapsack partitions.
- A320055Heinz numbers of sum-product knapsack partitions.
- A320054Number of spanning product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums the parts of a multiset partition of y is distinct.
- A320053Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.
- A320052Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.
- A319925Number of integer partitions with no 1's whose parts can be combined together using additions and multiplications to obtain n.
- A319916Number of integer partitions of any number from 1 to n whose product of parts is n.
- A319913Number of distinct integer partitions whose parts can be combined together using additions and multiplications to obtain n, with the exception that 1's can only be added and not multiplied with other parts.
- A319912Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n, such that m can be obtained by iteratively adding any two or multiplying any two non-1 parts of y until only one part (equal to m) remains.
- A319911Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n with no 1's, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.
- A319910Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.
- A319909Number of distinct positive integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with 1^n.
- A319907Number of distinct integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with the integer partition with Heinz number n.
- A319899Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).
- A319884Number of unordered pairs of set partitions of {1,...,n} where every block of one is a proper subset or proper superset of some block of the other.
- A319878Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).
- A319877Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).
- A319876Irregular triangle read by rows where T(n,k) is the number of permutations of {1,...,n} whose action on 2-element subsets of {1,...,n} has k cycles.
- A319856Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.
- A319855Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.
- A319850Number of distinct positive integers that can be obtained, starting with the initial interval partition (1, ..., n), by iteratively adding or multiplying together parts until only one part remains.
- A319841Number of distinct integers that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.
- A319837Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers.
- A319829FDH numbers of strict integer partitions of odd numbers.
- A319828FDH numbers of strict integer partitions of even numbers.
- A319827FDH numbers of relatively prime strict integer partitions.
- A319826GCD of the strict integer partition with FDH number n.
- A319825LCM of the strict integer partition with FDH number n.
- A319811Number of totally aperiodic integer partitions of n.
- A319810Number of fully periodic integer partitions of n.
- A319794Number of ways to split a strict integer partition of n into consecutive subsequences with weakly decreasing sums.
- A319793Number of non-isomorphic connected strict multiset partitions (sets of multisets) of weight n with empty intersection.
- A319792Number of non-isomorphic connected set systems of weight n with empty intersection.
- A319791Number of non-isomorphic connected set multipartitions (multisets of sets) of weight n with empty intersection.
- A319790Number of non-isomorphic connected multiset partitions of weight n with empty intersection.
- A319789Number of intersecting multiset partitions of strongly normal multisets of size n.
- A319787Number of intersecting multiset partitions of normal multisets of size n.
- A319786Number of factorizations of n where no two factors are relatively prime.
- A319784Number of non-isomorphic intersecting T_0 set systems of weight n.
- A319783Number of set systems spanning n vertices with empty intersection whose dual is also a set system with empty intersection.
- A319782Number of non-isomorphic intersecting strict T_0 multiset partitions of weight n.
- A319781Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.
- A319779Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.
- A319778Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.
- A319775Number of non-isomorphic multiset partitions of weight n with empty intersection and no part containing all the vertices.
- A319774Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.
- A319773Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.
- A319769Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.
- A319768Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n whose dual is a (not necessarily strict) intersecting multiset partition.
- A319767Number of non-isomorphic intersecting set systems spanning n vertices whose dual is also an intersecting set system.
- A319766Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n whose dual is also a strict intersecting multiset partition.
- A319765Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.
- A319764Number of non-isomorphic intersecting set systems of weight n with empty intersection.
- A319763Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.
- A319762Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.
- A319760Number of non-isomorphic intersecting strict multiset partitions (sets of multisets) of weight n.
- A319759Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.
- A319755Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n.
- A319752Number of non-isomorphic intersecting multiset partitions of weight n.
- A319751Number of non-isomorphic set systems of weight n with empty intersection.
- A319748Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.
- A319729Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.
- A319728Number of strict T_0 integer partitions of n.
- A319721Number of non-isomorphic antichains of multisets of weight n.
- A319719Number of non-isomorphic connected antichains of multisets of weight n.
- A319647Number of non-isomorphic connected set systems of weight n.
- A319646Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.
- A319645Number of non-isomorphic weight-n antichains of distinct multisets whose dual is a chain of distinct multisets.
- A319644Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.
- A319643Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.
- A319642Number of non-isomorphic weight-n antichains of distinct multisets whose dual is a chain of (not necessarily distinct) multisets.
- A319641Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.
- A319640Number of non-isomorphic antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.
- A319639Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.
- A319638Number of non-isomorphic weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.
- A319637Number of non-isomorphic T_0-covers of n vertices by distinct sets.
- A319635Number of non-isomorphic weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.
- A319634Number of non-isomorphic antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.
- A319633Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.
- A319632Number of non-isomorphic weight-n antichains of (not necessarily distinct) sets whose dual is also an antichain of (not necessarily distinct) sets.
- A319631Number of non-isomorphic weight-n antichains of multisets whose dual is a chain of distinct multisets.
- A319629Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.
- A319628Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.
- A319625Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.
- A319624Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.
- A319623Number of connected antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.
- A319622Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of (not necessarily distinct) sets.
- A319621Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of (not necessarily distinct) sets.
- A319620Number of connected antichain covers of n vertices by distinct sets whose dual is also a (not necessarily strict) antichain.
- A319619Number of non-isomorphic connected weight-n antichains of multisets whose dual is also an antichain of multisets.
- A319618Number of non-isomorphic weight-n antichains of multisets whose dual is a chain of multisets.
- A319616Number of non-isomorphic square multiset partitions of weight n.
- A319613a(n) = prime(n) * prime(2n).
- A319612Number of regular simple graphs spanning n vertices.
- A319567Product of y divided by the GCD of y to the power of the length of y, where y is the integer partition with Heinz number n.
- A319566Number of non-isomorphic connected T_0 set systems of weight n.
- A319565Number of non-isomorphic connected strict T_0 multiset partitions of weight n.
- A319564Number of non-isomorphic connected T_0 multiset partitions of weight n.
- A319560Number of non-isomorphic strict T_0 multiset partitions of weight n.
- A319559Number of non-isomorphic T_0 set systems of weight n.
- A319558The squarefree dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted without multiplicity. Then a(n) is the number of non-isomorphic multiset partitions of weight n whose squarefree dual is strict (no repeated blocks).
- A319557Number of non-isomorphic strict connected multiset partitions of weight n.
- A319540Number of unlabeled 3-uniform hypergraphs spanning n vertices such that every pair of vertices appears together in some block.
- A319496Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.
- A319437Number of series-reduced palindromic plane trees with n nodes.
- A319436Number of palindromic plane trees with n nodes.
- A319381Number of plane trees with n nodes where the sequence of branches directly under any given node is a membership chain.
- A319380Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of distinct multisets.
- A319379Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of multisets.
- A319378Number of plane trees with n nodes where the sequence of branches directly under any given node with at least two branches has empty intersection.
- A319336Denominator of the average of the averages of all integer partitions of n.
- A319335Numerator of the average of the averages of all integer partitions of n.
- A319334Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.
- A319333Heinz numbers of integer partitions whose sum is equal to their LCM.
- A319330Number of integer partitions of n whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts.
- A319329Heinz numbers of integer partitions whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts.
- A319328Heinz numbers of integer partitions such that not every distinct submultiset has a different GCD but every distinct submultiset has a different LCM.
- A319327Heinz numbers of integer partitions such that every distinct submultiset has a different LCM.
- A319320Number of integer partitions of n such that every distinct submultiset has a different LCM.
- A319319Heinz numbers of integer partitions such that every distinct submultiset has a different GCD.
- A319318Number of integer partitions of n such that every distinct submultiset has a different GCD.
- A319315Heinz numbers of integer partitions such that every distinct submultiset has a different average.
- A319312Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.
- A319301Sum of GCDs of strict integer partitions of n.
- A319300Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.
- A319299Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.
- A319292Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
- A319291Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.
- A319286Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
- A319285Number of series-reduced locally stable rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
- A319273Signed sum over the prime multiplicities of n.
- A319272Numbers whose prime multiplicities are distinct and whose prime indices are members of the sequence.
- A319271Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.
- A319270Numbers that are 1 or whose prime indices are relatively prime and belong to the sequence, and whose prime multiplicities are also relatively prime.
- A319269Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.
- A319255Number of strict antichains of multisets whose multiset union is an integer partition of n.
- A319247Irregular triangle whose n-th row lists the strict integer partition whose Heinz number is the n-th squarefree number.
- A319246Sum of prime indices of the n-th squarefree number.
- A319242Heinz numbers of strict integer partitions of odd numbers. Squarefree numbers whose prime indices sum to an odd number.
- A319241Heinz numbers of strict integer partitions of even numbers. Squarefree numbers whose prime indices sum to an even number.
- A319240Positions of zeroes in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).
- A319239Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).
- A319238Positions of zeroes in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).
- A319237Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).
- A319226Irregular triangle where T(n,k) is the number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the integer partition with Heinz number A215366(n,k).
- A319225Number of acyclic spanning subgraphs of a cycle graph, where the sizes of the connected components are given by the prime indices of n.
- A319193Irregular triangle where T(n,k) is the number of permutations of the integer partition with Heinz number A215366(n,k).
- A319192Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).
- A319191Coefficient of p(y) / A056239(n)! in Product_{i >= 1} (1 + x_i), where p is power-sum symmetric functions and y is the integer partition with Heinz number n.
- A319190Number of regular hypergraphs spanning n vertices.
- A319189Number of uniform regular hypergraphs spanning n vertices.
- A319187Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).
- A319182Irregular triangle where T(n,k) is the number of set partitions of {1,...,n} with block-sizes given by the integer partition with Heinz number A215366(n,k).
- A319181Numbers that are not perfect powers but whose prime indices have a common divisor > 1.
- A319180Perfect powers whose prime indices are relatively prime.
- A319179Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.
- A319169Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.
- A319165Perfect powers whose prime indices are not relatively prime.
- A319164Number of integer partitions of n that are neither relatively prime nor aperiodic.
- A319163Perfect powers whose prime multiplicities appear with relatively prime multiplicities.
- A319162Number of periodic integer partitions of n whose multiplicities are aperiodic, meaning the multiplicities of these multiplicities are relatively prime.
- A319161Numbers whose prime multiplicities appear with relatively prime multiplicities.
- A319160Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.
- A319157Smallest Heinz number of a superperiodic integer partition requiring n steps in the reduction to a multiset of size 1 obtained by repeatedly taking the multiset of multiplicities.
- A319153Number of integer partitions of n that reduce to 2, meaning their Heinz number maps to 2 under A304464.
- A319152Nonprime Heinz numbers of superperiodic integer partitions.
- A319151Heinz numbers of superperiodic integer partitions.
- A319149Number of superperiodic integer partitions of n.
- A319138Number of complete strict planar branching factorizations of n.
- A319137Number of strict planar branching factorizations of n.
- A319136Number of complete planar branching factorizations of n.
- A319123Number of series-reduced plane trees with n leaves such that each branch directly under any given node has a different number of leaves.
- A319122Number of phylogenetic plane trees on n labels.
- A319121Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.
- A319119Number of multimin tree-factorizations of Heinz numbers of integer partitions of n.
- A319118Number of multimin tree-factorizations of n.
- A319079Number of connected antichains of sets whose multiset union is an integer partition of n.
- A319077Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.
- A319071Number of integer partitions of n whose product of parts is a perfect power and whose parts all have the same number of prime factors, counted with multiplicity.
- A319066Number of partitions of integer partitions of n where all parts have the same length.
- A319058Number of relatively prime aperiodic factorizations of n into factors > 1.
- A319057Minimum sum of a strict factorization of n into factors > 1.
- A319056Number of non-isomorphic multiset partitions of weight n in which (1) all parts have the same size and (2) each vertex appears the same number of times.
- A319055Maximum product of an integer partition of n with relatively prime parts.
- A319054Maximum product of an aperiodic integer partition of n.
- A319005Number of integer partitions of n whose product of parts is >= n.
- A319004Number of ordered factorizations of n where the sequence of LCMs of the prime indices (A290103) of each factor is weakly increasing.
- A319003Number of ordered multiset partitions of integer partitions of n where the sequence of LCMs of the blocks is weakly increasing.
- A319002Number of ordered factorizations of n where the sequence of GCDs of prime indices (A289508) of the factors is weakly increasing.
- A319001Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.
- A319000Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.
- A318995Totally additive with a(prime(n)) = n - 1.
- A318994Totally additive with a(prime(n)) = n + 1.
- A318993Matula-Goebel number of the planted achiral tree determined by the n-th number whose consecutive prime indices are divisible.
- A318992Numbers whose consecutive prime indices are not all divisible.
- A318991Numbers whose consecutive prime indices are divisible. Heinz numbers of integer partitions in which each part is divisible by the next.
- A318990Numbers of the form prime(x) * prime(y) where x divides y.
- A318981Numbers whose prime indices plus 1 are relatively prime.
- A318980Number of integer partitions of n whose parts plus 1 are relatively prime.
- A318979Number of divisors of n with relatively prime prime indices, meaning they belong to A289508.
- A318978Heinz numbers of integer partitions with a common divisor > 1.
- A318954Minimum shifted Heinz number of a strict factorization of n into factors > 1.
- A318953Maximum Heinz number of a strict factorization of n into factors > 1.
- A318950Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.
- A318949Number of ways to write n as an orderless product of orderless sums.
- A318948Number of ways to choose an integer partition of each factor in a factorization of n.
- A318915Number of joining pairs of integer partitions of n.
- A318871Minimum Heinz number of a factorization of n into factors > 1.
- A318849Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.
- A318848Number of complete tree-partitions of a multiset whose multiplicities are the prime indices of n.
- A318847Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.
- A318846Number of total multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A318816Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.
- A318815Number of triples of set partitions of {1,2,...,n} whose join is { {1,...,n} }.
- A318813Number of total multiset partitions of 1^n. Number of total factorizations of prime^n.
- A318812Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.
- A318810Number of necklace permutations of a multiset whose multiplicities are the prime indices of n > 1.
- A318809Number of necklace permutations of the multiset of prime indices of n > 1.
- A318808Number of Lyndon permutations of a multiset whose multiplicities are the prime indices of n > 1.
- A318762Number of permutations of a multiset whose multiplicities are the prime indices of n.
- A318749Number of pairwise relatively nonprime strict factorizations of n (no two factors are coprime).
- A318748Number of compositions of n where adjacent parts are coprime and the last and first part are also coprime.
- A318747Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).
- A318746Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.
- A318745Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being coprime.
- A318731Number of relatively prime Lyndon compositions (aperiodic necklaces of positive integers) with sum n.
- A318730Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).
- A318729Number of cyclic compositions (necklaces of positive integers) summing to n with successive parts (including the last and first part) being indivisible.
- A318728Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being coprime.
- A318727Number of integer compositions of n where adjacent parts are indivisible (either way) and the last and first part are also indivisible (either way).
- A318726Number of integer compositions of n where successive parts are indivisible and the last and first part are also indivisible.
- A318721Number of strict relatively prime factorizations of n.
- A318720Numbers n such that there exists a strict relatively prime factorization of n in which no pair of factors is relatively prime.
- A318719Heinz numbers of strict integer partitions in which no two parts are relatively prime.
- A318718Heinz numbers of strict integer partitions with a common divisor > 1.
- A318717Number of strict integer partitions of n in which no two parts are relatively prime.
- A318716Heinz numbers of strict integer partitions with relatively prime parts in which no two parts are relatively prime.
- A318715Number of strict integer partitions of n with relatively prime parts in which no two parts are relatively prime.
- A318697Number of ways to partition a hypertree spanning n vertices into hypertrees.
- A318692Matula-Goebel numbers of series-reduced powerful uniform rooted trees.
- A318691Number of series-reduced powerful uniform rooted trees with n nodes.
- A318690Matula-Goebel numbers of powerful uniform rooted trees.
- A318689Number of powerful uniform rooted trees with n nodes.
- A318684Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.
- A318683Number of ways to split a strict integer partition of n into consecutive subsequences with equal sums.
- A318612Matula-Goebel numbers of powerful rooted trees.
- A318611Number of series-reduced powerful rooted trees with n nodes.
- A318589Heinz numbers of integer partitions whose sum of reciprocals squared is the reciprocal of an integer.
- A318588Heinz numbers of integer partitions whose sum of reciprocals squared is an integer.
- A318587Heinz numbers of integer partitions whose sum of reciprocals squared is 1.
- A318586Number of integer partitions of n whose sum of reciprocals squared is the reciprocal of an integer.
- A318585Number of integer partitions of n whose sum of reciprocals squared is an integer.
- A318584Number of integer partitions of n whose sum of reciprocals squared is 1.
- A318577Number of complete multimin tree-factorizations of n.
- A318574Denominator of the reciprocal sum of the integer partition with Heinz number n.
- A318573Numerator of the reciprocal sum of the integer partition with Heinz number n.
- A318567Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.
- A318566Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.
- A318565Number of multiset partitions of multiset partitions of strongly normal multisets of size n.
- A318564Number of multiset partitions of multiset partitions of normal multisets of size n.
- A318563Number of combinatory separations of strongly normal multisets of weight n.
- A318562Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.
- A318560Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.
- A318559Number of combinatory separations of the multiset of prime factors of n.
- A318532Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}} and join { {1,...,n} }.
- A318531Number of finite sets of set partitions of {1,...,n} such that any two have join { {1,...,n} }.
- A318485Number of p-trees of weight 2n + 1 in which all outdegrees are odd.
- A318434Number of ways to split the integer partition with Heinz number n into consecutive subsequences with equal sums.
- A318403Number of strict connected antichains of sets whose multiset union is an integer partition of n.
- A318402Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.
- A318401Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.
- A318400Numbers whose prime indices are all powers of 2 (including 1).
- A318399Number of triples of set partitions of {1,...,n} with meet {{1},...,{n}} and join { {1,...,n} }.
- A318398Number of triples of set partitions of {1,2,...,n} whose meet is {{1},{2},...,{n}}.
- A318396Number of pairs of integer partitions (y, v) of n such that there exists a pair of set partitions of {1,...,n} with meet {{1},...,{n}}, and with the first having block sizes y and the second v.
- A318395Number of nonnegative integer matrices with values summing to n, up to transposition and permutation of rows and columns.
- A318394Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}}.
- A318393Regular tetrangle where T(n,k,i) is the number of pairs of set partitions of {1,...,n} with meet of length k and join of length i.
- A318392Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join of length k.
- A318391Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.
- A318390Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join { {1,...,n} } and meet of length k.
- A318389Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet {{1},...,{n}} and join of length k.
- A318371Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.
- A318370Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.
- A318369Number of non-isomorphic set multipartitions (multisets of sets) of the multiset of prime indices of n.
- A318362Number of non-isomorphic set multipartitions of a multiset whose multiplicities are the prime indices of n.
- A318361Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.
- A318360Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.
- A318357Number of non-isomorphic strict multiset partitions of the multiset of prime indices of n.
- A318287Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A318286Number of strict multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A318285Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A318284Number of multiset partitions of a multiset whose multiplicities are the prime indices of n.
- A318283Sum of elements of the multiset spanning an initial interval of positive integers with multiplicities equal to the prime indices of n in weakly decreasing order.
- A318234Number of inequivalent leaf-colorings of totally transitive rooted trees with n nodes.
- A318231Number of inequivalent leaf-colorings of series-reduced rooted trees with n nodes.
- A318230Number of inequivalent leaf-colorings of binary rooted trees with 2n + 1 nodes.
- A318229Number of inequivalent leaf-colorings of transitive rooted trees with n nodes.
- A318228Number of inequivalent leaf-colorings of planted achiral trees with n nodes.
- A318227Number of inequivalent leaf-colorings of rooted identity trees with n nodes.
- A318226Number of inequivalent leaf-colorings of rooted trees with n nodes.
- A318187Number of totally transitive rooted trees with n leaves.
- A318186Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.
- A318185Number of totally transitive rooted trees with n nodes.
- A318153Number of antichain covers of the orderless Mathematica expression with e-number n.
- A318152e-numbers of unlabeled rooted trees. A number n is in the sequence iff n = 2^(prime(y_1) * ... * prime(y_k)) for some k > 0 and y_1, ..., y_k already in the sequence.
- A318151e-numbers of unlabeled rooted trees with with empty leaves o[] allowed.
- A318150e-numbers of free pure functions with one atom.
- A318149e-numbers of free pure symmetric multifunctions with one atom.
- A318120Number of set partitions of {1,...,n} with relatively prime block sizes.
- A318099Number of non-isomorphic weight-n antichains of (not necessarily distinct) multisets whose dual is also an antichain of (not necessarily distinct) multisets.
- A318049Number of first/rest balanced rooted plane trees with n nodes.
- A318048Size of the span of the unlabeled rooted tree with Matula-Goebel number n.
- A318046Number of initial subtrees (subtrees emanating from the root) of the unlabeled rooted tree with Matula-Goebel number n.
- A317994Number of inequivalent leaf-colorings of the unlabeled orderless Mathematica expression with e-number n.
- A317885Number of series-reduced free pure achiral multifunctions with one atom and n positions.
- A317884Number of series-reduced achiral Mathematica expressions with one atom and n positions.
- A317883Number of free pure achiral multifunctions with one atom and n positions.
- A317882Number of achiral Mathematica expressions with one atom and n positions.
- A317881Number of series-reduced identity Mathematica expressions with one atom and n positions.
- A317880Number of series-reduced free orderless identity Mathematica expressions with one atom and n positions.
- A317879Number of identity Mathematica expressions with one atom and n positions.
- A317878Number of free pure symmetric identity multifunctions with one atom and n positions.
- A317877Number of free pure identity multifunctions with one atom and n positions.
- A317876Number of free orderless identity Mathematica expressions with one atom and n positions.
- A317875Number of achiral free pure multifunctions with n unlabeled leaves.
- A317853a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).
- A317852Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.
- A317795Number of non-isomorphic set-systems spanning n vertices with no singletons.
- A317794Number of non-isomorphic set-systems on n vertices with no singletons.
- A317792Number of non-isomorphic multiset partitions, using normal multisets, of normal multisets of size n.
- A317791Number of non-isomorphic multiset partitions of the multiset of prime indices of n (row n of A112798).
- A317789Matula-Goebel numbers of rooted trees that are not locally nonintersecting.
- A317787Number of locally nonintersecting rooted trees with n nodes.
- A317786Matula-Goebel numbers of locally connected rooted trees.
- A317785Number of locally connected rooted trees with n nodes.
- A317776Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.
- A317775Number of strict multiset partitions of strongly normal multisets of size n, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
- A317765Number of distinct subexpressions of the orderless Mathematica expression with e-number n.
- A317757Number of non-isomorphic multiset partitions of size n such that the blocks have empty intersection.
- A317755Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.
- A317752Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.
- A317751Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.
- A317748Irregular triangle where T(n,d) is the number of factorizations of n into factors > 1 with GCD d.
- A317720Numbers that are not uniform relatively prime tree numbers.
- A317719Numbers that are not powerful tree numbers.
- A317718Number of uniform relatively prime rooted trees with n nodes.
- A317717Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.
- A317715Number of ways to split an integer partition of n into consecutive subsequences with equal sums.
- A317713Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.
- A317712Number of uniform rooted trees with n nodes.
- A317711Numbers that are not uniform tree numbers.
- A317710Uniform tree numbers. Matula-Goebel numbers of uniform rooted trees.
- A317709Aperiodic relatively prime tree numbers. Matula-Goebel numbers of aperiodic relatively prime trees.
- A317708Number of aperiodic relatively prime trees with n nodes.
- A317707Number of powerful rooted trees with n nodes.
- A317705Matula-Goebel numbers of series-reduced powerful rooted trees.
- A317677Fixed point of a shifted hypertree transform.
- A317676Triangle whose n-th row lists in order all e-numbers of orderless Mathematica expressions with one atom and n positions.
- A317674Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.
- A317672Regular triangle where T(n,k) is the number of clutters (connected antichains) on n + 1 vertices with k maximal blobs (2-connected components).
- A317671Regular triangle where T(n,k) is the number of labeled connected graphs on n + 1 vertices with k maximal blobs (2-connected components).
- A317659Regular triangle where T(n,k) is the number of distinct orderless Mathematica expressions with one atom, n positions, and k leaves.
- A317658Number of positions in the n-th orderless Mathematica expressions with one atom.
- A317656Number of free pure symmetric multifunctions whose leaves are the integer partition with Heinz number n.
- A317655Number of free pure symmetric multifunctions with leaves a multiset whose multiplicities are the integer partition with Heinz number n.
- A317654Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.
- A317653Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n.
- A317652Number of free pure symmetric multifunctions whose leaves are an integer partition of n.
- A317635Number of connected vertex sets of clutters (connected antichains) spanning n vertices.
- A317634Number of caps (also clutter partitions) of clutters (connected antichains) spanning n vertices.
- A317632Number of connected induced subgraphs of labeled connected graphs with n vertices.
- A317631Number of connected set partitions of the vertices of labeled graphs with n vertices.
- A317624Number of integer partitions of n where all parts are > 1 and whose LCM is n.
- A317616Numbers whose prime multiplicities are not pairwise indivisible.
- A317590Heinz numbers of integer partitions that are not uniformly normal.
- A317589Heinz numbers of uniformly normal integer partitions.
- A317588Number of uniformly normal integer partitions of n.
- A317584Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.
- A317583Number of multiset partitions of normal multisets of size n such that all blocks have the same size.
- A317581a(1) = 1; a(n > 1) = 1 + Sum_{d|n, d<n} mu(n/d) a(d).
- A317580Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.
- A317554Sum of coefficients in the expansion of p(y) in terms of Schur functions, where p is power-sum symmetric functions and y is the integer partition with Heinz number n.
- A317553Sum of coefficients in the expansion of Sum_{y a composition of n} p(y) in terms of Schur functions, where p is power-sum symmetric functions.
- A317552Irregular triangle where T(n,k) is the sum of coefficients in the expansion of p(y) in terms of Schur functions, where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).
- A317546Number of multimin partitions of integer partitions of n.
- A317545Number of multimin factorizations of n.
- A317534Numbers n such that the poset of factorizations of n, ordered by refinement, is not a lattice.
- A317533Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of size n and length k.
- A317532Regular triangle where T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.
- A317508Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.
- A317493Heinz numbers of integer partitions that are not fully normal.
- A317492Heinz numbers of fully normal integer partitions.
- A317491Number of fully normal integer partitions of n.
- A317449Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
- A317258Heinz numbers of integer partitions that are not totally nonincreasing.
- A317257Heinz numbers of totally nonincreasing integer partitions.
- A317256Number of totally nonincreasing integer partitions of n.
- A317246Heinz numbers of supernormal integer partitions.
- A317245Number of supernormal integer partitions of n.
- A317176Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).
- A317146Moebius function in the ranked poset of factorizations of n into factors > 1, evaluated at the minimum (the prime factorization of n).
- A317145Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.
- A317144Number of refinement-ordered pairs of factorizations of n into factors > 1.
- A317143In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.
- A317142Number of refinement-ordered pairs of strict integer partitions of n.
- A317141In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition with Heinz number n.
- A317102Powerful numbers whose prime multiplicities are pairwise indivisible.
- A317101Numbers whose prime multiplicities are pairwise indivisible.
- A317100Number of series-reduced planted achiral trees with n leaves spanning an initial interval of positive integers.
- A317099Number of series-reduced planted achiral trees whose leaves span an initial interval of positive integers appearing with multiplicities an integer partition of n.
- A317098Number of series-reduced rooted trees with n unlabeled leaves where the number of distinct branches under each node is <= 2.
- A317097Number of rooted trees with n nodes where the number of distinct branches under each node is <= 2.
- A317092Positive integers whose prime multiplicities are weakly decreasing and span an initial interval of positive integers.
- A317091Positive integers whose prime multiplicities are weakly increasing and span an initial interval of positive integers.
- A317090Positive integers whose prime multiplicities span an initial interval of positive integers.
- A317089Numbers whose prime factors span an initial interval of prime numbers and whose prime multiplicities span an initial interval of positive integers.
- A317088Number of normal integer partitions of n whose multiset of multiplicities is also normal.
- A317087Numbers whose prime factors span an initial interval of prime numbers and whose sequence of prime multiplicities is a palindrome.
- A317086Number of normal integer partitions of n whose sequence of multiplicities is a palindrome.
- A317085Number of integer partitions of n whose sequence of multiplicities is a palindrome.
- A317084Number of integer partitions of n whose multiplicities are weakly increasing and span an initial interval of positive integers.
- A317082Number of integer partitions of n whose multiplicities are weakly decreasing and span an initial interval of positive integers.
- A317081Number of integer partitions of n whose multiplicities span an initial interval of positive integers.
- A317080Number of unlabeled connected antichains of multisets with multiset-join a multiset of size n.
- A317079Number of unlabeled antichains of multisets with multiset-join a multiset of size n.
- A317078Number of connected multiset partitions of strongly normal multisets of size n.
- A317077Number of connected multiset partitions of normal multisets of size n.
- A317076Number of connected antichains of multisets with multiset-join a strongly normal multiset of size n.
- A317075Number of connected antichains of multisets with multiset-join a normal multiset of size n.
- A317074Number of antichains of multisets with multiset-join a strongly normal multiset of size n.
- A317073Number of antichains of multisets with multiset-join a normal multiset of size n.
- A317056Depth of the orderless Mathematica expression with e-number n.
- A316983Number of non-isomorphic self-dual multiset partitions of weight n.
- A316981Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.
- A316980Number of non-isomorphic strict multiset partitions of weight n.
- A316979Number of strict factorizations of n into factors > 1 with no equivalent primes.
- A316978Number of factorizations of n into factors > 1 with no equivalent primes.
- A316977Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.
- A316974Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.
- A316972Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.
- A316904Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is an integer.
- A316903Heinz numbers of aperiodic integer partitions whose reciprocal sum is the reciprocal of an integer.
- A316902Heinz numbers of aperiodic integer partitions whose reciprocal sum is an integer.
- A316901Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is the reciprocal of an integer.
- A316900Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is an integer.
- A316899Number of integer partitions of n into relatively prime parts whose reciprocal sum is an integer.
- A316898Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.
- A316897Number of integer partitions of n into relatively prime parts whose reciprocal sum is 1.
- A316896Number of aperiodic integer partitions of n whose reciprocal sum is 1.
- A316895Number of aperiodic integer partitions of n whose reciprocal sum is an integer.
- A316894Number of aperiodic integer partitions of n whose reciprocal sum is the reciprocal of an integer.
- A316893Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is 1.
- A316892Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices.
- A316891Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is an integer.
- A316890Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is 1.
- A316889Heinz numbers of aperiodic integer partitions whose reciprocal sum is 1.
- A316888Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.
- A316857Heinz numbers of integer partitions whose reciprocal sum is the reciprocal of an integer.
- A316856Heinz numbers of integer partitions whose reciprocal sum is an integer.
- A316855Heinz numbers of integer partitions whose reciprocal sum is 1.
- A316854Number of integer partitions of n whose reciprocal sum is the reciprocal of an integer.
- A316796Number of unlabeled rooted trees with n nodes in which all multiplicities of branches under any given node are distinct.
- A316795Number of aperiodic rooted trees on n nodes with locally distinct multiplicities.
- A316794Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.
- A316793Numbers whose prime multiplicities are distinct and relatively prime.
- A316790Number of orderless same-tree-factorizations of n.
- A316789Number of same-tree-factorizations of n.
- A316784Number of orderless identity tree-factorizations of n.
- A316782Number of achiral tree-factorizations of n.
- A316772Number of series-reduced locally nonintersecting rooted trees whose leaves form an integer partition of n.
- A316771Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number n.
- A316770Number of series-reduced locally nonintersecting rooted identity trees whose leaves form an integer partition of n.
- A316769Number of series-reduced locally stable rooted trees with n unlabeled leaves.
- A316768Number of series-reduced locally stable rooted trees whose leaves form an integer partition of n.
- A316767Number of series-reduced locally stable rooted trees whose leaves form the integer partition with Heinz number n.
- A316766Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.
- A316697Number of series-reduced locally disjoint rooted trees with n unlabeled leaves.
- A316696Number of series-reduced locally disjoint rooted trees whose leaves form an integer partition of n.
- A316695Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.
- A316694Number of series-reduced locally disjoint rooted identity trees whose leaves form an integer partition of n.
- A316656Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.
- A316655Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.
- A316654Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
- A316653Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.
- A316652Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
- A316651Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers.
- A316624Number of balanced p-trees with n leaves.
- A316597Heinz numbers of non-totally nondecreasing integer partitions.
- A316557Number of distinct integer averages of subsets of the integer partition with Heinz number n.
- A316556Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.
- A316555Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.
- A316529Heinz numbers of totally nondecreasing integer partitions.
- A316525Numbers whose average of prime factors is prime.
- A316524Signed sum over the prime indices of n.
- A316523Number of odd multiplicities minus number of even multiplicities in the canonical prime factorization of n.
- A316522Number of unlabeled rooted trees with n nodes where all terminal rooted subtrees are either constant or strict.
- A316521Matula-Goebel numbers of rooted trees where all terminal rooted subtrees are either constant or strict.
- A316520Heinz numbers of integer partitions whose average is a prime number.
- A316503Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.
- A316502Matula-Goebel numbers of unlabeled rooted trees with n nodes in which the branches of any node with more than one branch have empty intersection.
- A316501Number of unlabeled rooted trees with n nodes in which the branches of any node with more than one distinct branch have empty intersection.
- A316500Number of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.
- A316496Number of totally nondecreasing integer partitions of n.
- A316495Matula-Goebel numbers of locally disjoint rooted trees, meaning no branch overlaps any other (unequal) branch of the same root.
- A316494Matula-Goebel numbers of locally disjoint rooted identity trees, meaning no branch overlaps any other branch of the same root.
- A316476Stable numbers. Numbers whose distinct prime indices are pairwise indivisible.
- A316475Number of locally stable rooted trees with n nodes, meaning no branch is a submultiset of any other (unequal) branch of the same root.
- A316474Number of locally stable rooted identity trees with n nodes, meaning no branch is a subset of any other branch of the same root.
- A316473Number of locally disjoint rooted trees with n nodes, meaning no branch overlaps any other (unequal) branch of the same root.
- A316471Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.
- A316470Matula-Goebel numbers of unlabeled rooted RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.
- A316469Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.
- A316468Matula-Goebel numbers of locally stable rooted trees, meaning no branch is a submultiset of any other branch of the same root.
- A316467Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.
- A316465Heinz numbers of integer partitions such that every nonempty submultiset has an integer average.
- A316441a(n) = Sum (-1)^k where the sum is over all factorizations of n into factors > 1 and k is the number of factors.
- A316440Number of integer partitions of n such that every submultiset has an integer average.
- A316439Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).
- A316438Heinz numbers of integer partitions whose product is strictly greater than their lcm.
- A316437Take the integer partition with Heinz number n, divide all parts by the gcd of the parts, then take the Heinz number of the resulting partition.
- A316436Sum divided by gcd of the integer partition with Heinz number n > 1.
- A316433Number of integer partitions of n whose length is equal to their lcm.
- A316432Number of integer partitions of n whose length is equal to their gcd.
- A316431Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.
- A316430Heinz numbers of integer partitions whose length is equal to their gcd.
- A316429Heinz numbers of integer partitions whose length is equal to their lcm.
- A316428Heinz numbers of integer partitions such that every part is divisible by the number of parts.
- A316413Heinz numbers of integer partitions whose length divides their sum.
- A316402Number of strict non-knapsack integer partitions of n, meaning not every subset has a different sum.
- A316401Number of strict integer partitions of n that are not knapsack (not every subset has a different sum) but every subset has a different average.
- A316400Number of strict integer partitions of n that are knapsack (every subset has a different sum) but not every subset has a different average.
- A316399Number of strict integer partitions of n such that not every subset has a different average.
- A316398Number of distinct subset-averages of the integer partition with Heinz number n.
- A316365Number of factorizations of n into factors > 1 such that every distinct subset of the factors has a different sum.
- A316364Number of factorizations of n into factors > 1 such that every distinct submultiset of the factors has a different average.
- A316362Heinz numbers of strict integer partitions such that not every distinct subset has a different average.
- A316361FDH numbers of strict integer partitions such that not every distinct subset has a different average.
- A316314Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.
- A316313Number of integer partitions of n such that every distinct submultiset has a different average.
- A316271FDH numbers of strict non-knapsack partitions.
- A316268FDH numbers of connected strict integer partitions.
- A316267FDH numbers of strict integer partitions of prime numbers with a prime number of prime parts.
- A316266FDH numbers of strict integer partitions with prime parts and prime length.
- A316265FDH numbers of strict integer partitions with prime parts.
- A316264FDH numbers of strict integer partitions with odd length and all odd parts.
- A316245Number of ways to split an integer partition of n into consecutive subsequences with weakly decreasing sums.
- A316228Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.
- A316223Number of subset-sum triangles with composite a subset-sum of the integer partition with Heinz number n.
- A316222Number of positive subset-sum triangles whose composite is a positive subset-sum of an integer partition of n.
- A316220Number of triangles of weight the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.
- A316219Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.
- A316211Number of strict integer partitions of n into Fermi-Dirac primes.
- A316210Number of integer partitions of the n-th Fermi-Dirac prime into Fermi-Dirac primes.
- A316202Number of integer partitions of n into Fermi-Dirac primes.
- A316185Number of strict integer partitions of the n-th prime into a prime number of prime parts.
- A316154Number of integer partitions of prime(n) into a prime number of prime parts.
- A316153Heinz numbers of integer partitions of prime numbers into a prime number of prime parts.
- A316151Heinz numbers of strict integer partitions of prime numbers into prime parts.
- A316112Number of leaves in the orderless Mathematica expression with e-number n.
- A316094FDH numbers of strict integer partitions with odd parts.
- A316092Heinz numbers of integer partitions of prime numbers into prime parts.
- A316091Heinz numbers of integer partitions of prime numbers.
- A309667Number of non-isomorphic connected set-systems on up to n vertices.
- A309615Number of T_0 set-systems covering n vertices that are closed under intersection.
- A309356MM-numbers of labeled simple covering graphs.
- A309326BII-numbers of minimal covers.
- A309314BII-numbers of hyperforests.
- A308546Number of double-closed subsets of {1...n}.
- A308542Number of subsets of {2...n} containing the product of any set of distinct elements whose product is <= n.
- A308299Numbers whose prime indices are factorial numbers.
- A308251Number of subsets of {1,...,n + 1} containing n + 1 and such that all absolute value differences of distinct elements are distinct.
- A307895Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.
- A307824Heinz numbers of integer partitions whose augmented differences are all equal.
- A307734Smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k.
- A307699Numbers n such that there is no integer partition of n with exactly n - 1 submultisets.
- A307539Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
- A307534Heinz numbers of strict integer partitions with 3 parts, all of which are odd.
- A307517Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.
- A307516Numbers whose maximum prime index and minimum prime index differ by more than 1.
- A307515Heinz numbers of integer partitions with Durfee square of length > 2.
- A307386Heinz numbers of integer partitions with Durfee square of length 3.
- A307373Heinz numbers of integer partitions with at least three parts, the third of which is 2.
- A307370Number of integer partitions of n with 2 distinct parts, none appearing more than twice.
- A307249Number of simplicial complexes with n nodes.
- A307230Number of divisible pairs of distinct positive integers up to n with at least one binary carry.
- A306844Number of anti-transitive rooted trees with n nodes.
- A306719Lexicographically earliest sequence containing 2 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.
- A306715Number of graphical necklaces with n vertices and distinct rotations.
- A306669Number of aperiodic permutation necklaces of weight n.
- A306558Number of double-crossing set partitions of {1, ..., n}.
- A306551Number of non-double-crossing set partitions of {1,...,n}.
- A306550Array read by antidiagonals where A(n,k) is the number of labeled k-antichains covering n vertices.
- A306505Number of non-isomorphic antichains of nonempty sets.
- A306438Number of non-crossing set partitions whose block sizes are the prime indices of n.
- A306437Regular triangle read by rows where T(n, k) is the number of non-crossing set partitions of {1, ..., n} in which all blocks have size k.
- A306419Number of set partitions of {1, ..., n} whose blocks are all singletons and pairs, not including {1, n} or {i, i + 1} for any i.
- A306418Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} requiring k steps of removing singletons and cyclical adjacency initiators until reaching a fixed point, n >= 0, 0 <= k <= n.
- A306417Number of self-conjugate set partitions of {1, ..., n}.
- A306416Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).
- A306386Number of chord diagrams with n chords all having arc length at least 3.
- A306357Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.
- A306351Number of ways to split an n-cycle into connected subgraphs all having at least 4 vertices.
- A306320Number of square plane partitions of n with distinct row sums and distinct column sums.
- A306319Number of rectangular twice-partitions of n.
- A306318Number of square twice-partitions of n.
- A306269Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.
- A306268Number of ways to choose a strict factorization into squarefree factors of each factor in a strict factorization of n.
- A306203Matula-Goebel numbers of balanced rooted semi-identity trees.
- A306202Matula-Goebel numbers of rooted semi-identity trees.
- A306201Number of unlabeled balanced rooted semi-identity trees with n nodes.
- A306200Number of unlabeled rooted semi-identity trees with n nodes.
- A306186Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.
- A306021Number of set-systems spanning n vertices in which all parts have the same size.
- A306020Number of set-systems using nonempty subsets of {1,...,n} in which all sets have the same size.
- A306019Number of non-isomorphic set-systems of weight n in which all parts have the same size.
- A306018Number of non-isomorphic set multipartitions of weight n in which all parts have the same size.
- A306017Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.
- A306008Number of non-isomorphic intersecting set-systems of weight n with no singletons.
- A306007Number of non-isomorphic intersecting antichains of weight n.
- A306006Number of non-isomorphic intersecting set-systems of weight n.
- A306005Number of non-isomorphic set-systems of weight n with no singletons.
- A306001Number of unlabeled intersecting set-systems with no singletons on up to n vertices.
- A306000Number of labeled intersecting set-systems with no singletons covering some subset of {1,...,n}.
- A305999Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.
- A305940Irregular triangle where T(n,k) is the coefficient of s(y) in p(n), where s is Schur functions, p is power-sum symmetric functions, and y is the integer partition with Heinz number A215366(n,k).
- A305936Irregular triangle whose n-th row is the multiset spanning an initial interval of positive integers with multiplicities equal to the n-th row of A296150 (the prime indices of n in weakly decreasing order).
- A305935Number of labeled spanning intersecting set-systems on n vertices with no singletons.
- A305857Number of unlabeled intersecting antichains on up to n vertices.
- A305856Number of unlabeled intersecting set-systems on up to n vertices.
- A305855Number of unlabeled spanning intersecting antichains on n vertices.
- A305854Number of unlabeled spanning intersecting set-systems on n vertices.
- A305844Number of labeled spanning intersecting antichains on n vertices.
- A305843Number of labeled spanning intersecting set-systems on n vertices.
- A305832Number of connected components of the n-th FDH set-system.
- A305831Number of connected components of the strict integer partition with FDH number n.
- A305830Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).
- A305829Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.
- A305761Nonprime Heinz numbers of z-trees.
- A305736Number of integer partitions of n whose greatest common divisor is composite (nonprime and > 1).
- A305735Number of integer partitions of n whose greatest common divisor is a prime number.
- A305733Heinz numbers of irreducible integer partitions. Nonprime numbers whose prime indices have a common divisor > 1 or such that A181819(n) is already in the sequence.
- A305732Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.
- A305731Number of irreducible integer partitions of n.
- A305715Irregular triangle whose rows are all finite sequences of positive integers that are polydivisible and strictly pandigital.
- A305714Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.
- A305713Number of strict integer partitions of n into pairwise coprime parts.
- A305712Polydivisible nonnegative integers whose decimal digits span an initial interval of {0,...,9}.
- A305701Nonnegative integers whose decimal digits span an initial interval of {0,...,9}.
- A305634Even numbers that are not perfect powers.
- A305633Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).
- A305632Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).
- A305631Expansion of Product_{r not a perfect power} 1/(1 - x^r).
- A305630Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).
- A305614Expansion of Sum_{p prime} x^p/(1 + x^p).
- A305613Numbers whose multiset of prime factors is not knapsack.
- A305611Number of distinct positive subset-sums of the multiset of prime factors of n.
- A305610Signed recurrence over orderless same-trees: a(n) = (-1)^(n-1) + Sum_{d|n, d>1} binomial(a(n/d) + d - 1, d).
- A305572Signed recurrence over same-trees: a(1) = 1; a(n > 1) = (-1)^(n-1) + Sum_{d|n, d>1} a(n/d)^d.
- A305567Irregular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k, where k runs over all divisors of n.
- A305566Number of finite sets of relatively prime positive integers > 1 with least common multiple n.
- A305565Regular triangle where T(n,k) is the number of finite sets of positive integers with least common multiple n and greatest common divisor k.
- A305564Number of finite sets of relatively prime positive integers with least common multiple n.
- A305563Number of reducible integer partitions of n.
- A305552Number of uniform normal multiset partitions of weight n.
- A305551Number of partitions of partitions of n where all partitions have the same sum.
- A305504Heinz numbers of integer partitions whose distinct parts plus 1 are connected.
- A305501Number of connected components of the integer partition y + 1 where y is the integer partition with Heinz number n.
- A305254Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.
- A305253Number of connected, pairwise indivisible factorizations of n into factors greater than 1.
- A305195Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.
- A305194Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.
- A305193Number of connected factorizations of n.
- A305150Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.
- A305149Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.
- A305148Number of integer partitions of n whose distinct parts are pairwise indivisible.
- A305106Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.
- A305103Heinz numbers of connected integer partitions with z-density -1.
- A305081Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.
- A305080Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.
- A305079Number of connected components of the integer partition with Heinz number n.
- A305078Heinz numbers of connected integer partitions.
- A305055Numbers n such that the z-density of the integer partition with Heinz number n is 0.
- A305054If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i), where omega = A001221 is number of distinct prime factors.
- A305053If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i) - omega(n), where omega = A001221 is number of distinct prime factors.
- A305052z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).
- A305028Number of unlabeled blobs spanning n vertices without singleton edges.
- A305005Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} without singleton edges.
- A305004Number of labeled hypertrees (connected acyclic antichains) spanning some subset of {1,...,n} without singleton edges.
- A305001Number of labeled antichains of finite sets spanning n vertices without singletons.
- A305000Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.
- A304999Number of labeled antichains of finite sets spanning n vertices with singleton edges allowed.
- A304998Number of unlabeled antichains of finite sets spanning n vertices without singletons.
- A304997Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.
- A304996Number of unlabeled antichains of finite sets spanning up to n vertices with singleton edges allowed.
- A304986Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form are allowed for any vertex x.
- A304985Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.
- A304984Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.
- A304983Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.
- A304982Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.
- A304981Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.
- A304977Number of unlabeled hyperforests spanning n vertices with singleton edges allowed.
- A304970Number of unlabeled hypertrees with up to n vertices and without singleton edges.
- A304968Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.
- A304939Number of labeled nonempty hypertrees (connected acyclic antichains) spanning some subset of {1,...,n} without singleton edges.
- A304937Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.
- A304919Number of labeled hyperforests spanning {1,...,n} and allowing singleton edges.
- A304918Number of labeled antichain hyperforests spanning a subset of {1,...,n}.
- A304912Number of non-isomorphic spanning hyperforests of weight n.
- A304911Number of labeled hyperforests spanning n vertices without singleton edges.
- A304887Number of non-isomorphic blobs of weight n.
- A304867Number of non-isomorphic hypertrees of weight n.
- A304820The rootless co-delta function. Dirichlet inverse of the rootless delta function A304819.
- A304819The rootless delta function. Dirichlet convolution of r with zeta, where r(n) = (-1)^Omega(n) if n is 1 or not a perfect power (rootless) and r(n) = 0 otherwise.
- A304818a(Product_i prime(y_i)) = Sum_i y_i*i.
- A304817Number of divisors of n that are either 1 or not a perfect power.
- A304796Number of special sums of integer partitions of n.
- A304795Number of positive special sums of the integer partition with Heinz number n.
- A304793Number of distinct positive subset-sums of the integer partition with Heinz number n.
- A304792Number of subset-sums of integer partitions of n.
- A304779The rootless zeta function. Dirichlet inverse of the rootless Moebius function defined by r(n) = (-1)^Omega(n) if n is 1 or not a perfect power (rootless) and r(n) = 0 otherwise.
- A304776A weakening function.
- A304768Augmented integer conjugate of n.
- A304717Number of connected strict integer partitions of n with pairwise indivisible parts.
- A304716Number of connected integer partitions of n.
- A304714Number of connected strict integer partitions of n.
- A304713Squarefree numbers whose prime indices are pairwise indivisible. Heinz numbers of strict integer partitions with pairwise indivisible parts.
- A304712Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.
- A304711Heinz numbers of integer partitions whose distinct parts are pairwise coprime.
- A304709Number of integer partitions of n whose distinct parts are pairwise coprime.
- A304687Prime Omicron (big-O) of n.
- A304686Numbers with strictly decreasing prime multiplicities.
- A304679A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).
- A304678Numbers with weakly increasing prime multiplicities.
- A304660A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).
- A304653The rootless Moebius function. r(n) = (-1)^Omega(n) if n is 1 or not a perfect power (rootless) and r(n) = 0 otherwise.
- A304650Number of ways to write n as a product of two numbers, neither of which is a perfect power.
- A304649Number of divisors d|n such that neither d nor n/d is a perfect power greater than 1.
- A304648Number of different periodic multisets that fit within some normal multiset of weight n.
- A304647Smallest member of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.
- A304636Numbers n with prime omicron 3, meaning A304465(n) = 3.
- A304634Numbers n with prime omicron 2, meaning A304465(n) = 2.
- A304623Regular triangle where T(n,k) is the number of aperiodic multisets with maximum k that fit within some normal multiset of weight n.
- A304576a(n) = Sum_{k < n, k squarefree and relatively prime to n} (-1)^(k-1).
- A304575a(n) = Sum_{d|n} #{k < d, k squarefree and relatively prime to d}.
- A304574Number of perfect powers (A001597) less than n and relatively prime to n.
- A304573Number of non-perfect powers (A007916) less than n and relatively prime to n.
- A304495Decapitate the power-tower for n, i.e., remove the last or deepest exponent.
- A304492Position in the sequence of numbers that are not perfect powers (A007916) of the last or deepest exponent in the power-tower for n.
- A304491Last or deepest exponent in the power-tower for n.
- A304486Number of inequivalent leaf-colorings of the unlabeled rooted tree with Matula-Goebel number n.
- A304485Regular triangle where T(n,k) is the number of distinct unlabeled orderless Mathematica expressions with n positions and k leaves.
- A304481Turn the power-tower for n upside-down.
- A304465Prime omicron of n.
- A304464Start with the normalized multiset of prime factors of n > 1. Given a multiset, take the multiset of its multiplicities. Repeat this until a multiset of size 1 is obtained. a(n) is the unique element of this multiset.
- A304455Number of steps in the reduction to a multiset of size 1 of the multiset of prime factors of n, obtained by repeatedly taking the multiset of multiplicities.
- A304450Numbers that are not perfect powers and whose prime factors span an initial interval of prime numbers.
- A304449Numbers that are either squarefree or a perfect power.
- A304438Coefficient of s(y) in p(|y|), where s is Schur functions, p is power-sum symmetric functions, y is the integer partition with Heinz number n, and |y| = Sum y_i.
- A304386Number of unlabeled hypertrees spanning up to n vertices with singleton edges allowed.
- A304382Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.
- A304369Numbers n such that Sum_{d|n, d = 1 or not a perfect power} mu(n/d) is greater than 1 in absolute value.
- A304365Numbers n such that Sum_{d|n, d = 1 or not a perfect power} mu(n/d) is nonzero.
- A304364Numbers n such that A304362(n) = Sum_{d|n, d = 1 or not a perfect power} mu(n/d) = 0.
- A304362a(n) = Sum_{d|n, d = 1 or not a perfect power} mu(n/d).
- A304360Numbers > 1 whose prime indices are not in the sequence.
- A304339Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).
- A304328a(n) = n/(largest perfect power divisor of n).
- A304327Number of ways to write n as a product of a perfect power and a squarefree number.
- A304326Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.
- A304250Perfect powers whose prime factors span an initial interval of prime numbers.
- A304175Number of leaf-balanced rooted plane trees with n nodes.
- A304173Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.
- A304118Number of z-blobs with least common multiple n > 1.
- A303976Number of different aperiodic multisets that fit within some normal multiset of size n.
- A303975Number of distinct prime factors in the product of prime indices of n.
- A303974Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.
- A303946Numbers that are neither squarefree nor perfect powers.
- A303945Triangle whose n-th row lists the multiset of prime indices of the n-th number that is not a perfect power A007916(n).
- A303838Number of z-forests with least common multiple n > 1.
- A303837Number of z-trees with least common multiple n > 1.
- A303710Number of factorizations of A007916(n) using elements of A007916 (rootless numbers).
- A303709Number of periodic factorizations of n using elements of A007916 (rootless numbers).
- A303708Number of aperiodic factorizations of n using elements of A007916 (rootless numbers).
- A303707Number of factorizations of n using elements of A007916 (rootless numbers).
- A303674Number of connected integer partitions of n > 1 whose distinct parts are pairwise indivisible and whose z-density is -1.
- A303554Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.
- A303553Number of periodic factorizations of n > 1 into positive factors greater than 1.
- A303552Number of periodic multisets of compositions of total weight n.
- A303551Number of aperiodic multisets of compositions of total weight n.
- A303547Number of non-isomorphic periodic multiset partitions of weight n.
- A303546Number of non-isomorphic aperiodic multiset partitions of weight n.
- A303431Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.
- A303386Number of aperiodic factorizations of n > 1.
- A303365Number of integer partitions of the n-th squarefree number using squarefree numbers.
- A303364Number of strict integer partitions of n with pairwise indivisible and squarefree parts.
- A303362Number of strict integer partitions of n with pairwise indivisible parts.
- A303283Squarefree numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.
- A303282Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.
- A303280Number of strict integer partitions of n whose parts have a common divisor other than 1.
- A303140Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.
- A303139Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.
- A303138Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.
- A303027Number of orderless Mathematica expressions with no empty or unitary parts (subexpressions of the form x[] or x[y] where x and y are both orderless Mathematica expressions).
- A303026Matula-Goebel numbers of series-reduced anti-binary (no unary or binary branchings) rooted trees.
- A303025Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.
- A303024Matula-Goebel numbers of anti-binary (no binary branchings) rooted trees.
- A303023Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.
- A303022Number of orderless Mathematica expressions with one atom, n positions, and no unitary parts (subexpressions of the form x[y] where x and y are both orderless Mathematica expressions).
- A302979Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.
- A302917Solution to a(1) = 1 and Sum_y Product_i a(y_i) = 0 for each n > 1, where the sum is over all relatively prime or monic partitions of n.
- A302916Number of relatively prime p-trees of weight n.
- A302915Number of relatively prime enriched p-trees of weight n.
- A302798Squarefree numbers that are prime or whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions that either consist of a single part or have pairwise coprime parts.
- A302797Squarefree numbers whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions with pairwise coprime parts.
- A302796Squarefree numbers whose prime indices are relatively prime. Nonprime Heinz numbers of strict integer partitions with relatively prime parts.
- A302698Number of integer partitions of n into relatively prime parts that are all greater than 1.
- A302697Odd numbers whose prime indices are relatively prime. Heinz numbers of integer partitions with no 1s and with relatively prime parts.
- A302696Numbers whose prime indices are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.
- A302602Numbers that are powers of a prime number whose prime index is either 1 or a prime number.
- A302601Numbers that are powers of a prime number whose prime index is also a prime power (not including 1).
- A3026001, 2, prime numbers of prime index, and twice prime numbers of prime index.
- A302597Squarefree numbers whose prime indices are powers of a common prime number.
- A302596Powers of prime numbers of prime index.
- A302594Numbers whose prime indices other than 1 are equal prime numbers.
- A302593Numbers whose prime indices are powers of a common prime number.
- A302592One, powers of 2, and prime numbers of prime index.
- A302591One, powers of 2, and prime numbers of squarefree index.
- A302590Squarefree numbers whose prime indices are prime numbers. Numbers that are a product of distinct prime numbers of prime index. Products of distinct prime numbers of prime index.
- A302569Numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts.
- A302568Odd numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts all greater than 1.
- A302546a(n) = Sum_{d = 1...n} 2^binomial(n, d).
- A302545Number of non-isomorphic multiset partitions of weight n with no singletons.
- A302540Numbers whose prime indices other than 1 are prime numbers.
- A302539Squarefree numbers whose prime indices other than 1 are prime numbers.
- A302534Squarefree numbers whose prime indices are also squarefree and have disjoint prime indices.
- A302521Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.
- A302505Numbers whose prime indices are squarefree and have disjoint prime indices.
- A302498Numbers that are a power of a prime number whose prime index is itself a power of a prime number.
- A302497Powers of primes of squarefree index.
- A302496Products of distinct primes of prime-power index.
- A302494Products of distinct primes of squarefree index.
- A302493Prime numbers of prime-power index.
- A302492Products of any power of 2 with prime numbers of prime-power index, i.e. prime numbers p of the form p = prime(q^k), for q prime, k >= 1.
- A302491Prime numbers of squarefree index.
- A302478Products of prime numbers of squarefree index.
- A302394Number of families of 3-subsets of an n-set that cover all 2-subsets.
- A302243Total weight of the n-th twice-odd-factored multiset partition.
- A302242Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).
- A302129Number of unlabeled uniform connected hypergraphs of weight n.
- A302094Number of relatively prime or monic twice-partitions of n.
- A301988Nonprime Heinz numbers of integer partitions whose product is equal to their sum.
- A301987Heinz numbers of integer partitions whose product is equal to their sum.
- A301979Number of subset-sums minus number of subset-products of the integer partition with Heinz number n.
- A301970Heinz numbers of integer partitions with more subset-products than subset-sums.
- A301957Number of distinct subset-products of the integer partition with Heinz number n.
- A301935Number of positive subset-sum trees whose composite a positive subset-sum of the integer partition with Heinz number n.
- A301934Number of positive subset-sum trees of weight n.
- A301924Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.
- A301922Regular triangle where T(n,k) is the number of unlabeled k-uniform hypergraphs spanning n vertices.
- A301920Number of unlabeled uniform connected hypergraphs spanning n vertices.
- A301900Heinz numbers of strict non-knapsack partitions. Squarefree numbers such that multiple divisors have the same Heinz weight A056239(d).
- A301899Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).
- A301856Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.
- A301855Number of divisors d|n such that no other divisor of n has the same Heinz weight A056239(d).
- A301854Number of positive special sums of integer partitions of n.
- A301830Number of factorizations of n into factors (greater than 1) of two kinds.
- A301829Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.
- A301768Number of ways to choose a strict rooted partition of each part in a constant rooted partition of n.
- A301767Number of ways to choose a constant rooted partition of each part in a strict rooted partition of n.
- A301766Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).
- A301765Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).
- A301764Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n such that the flattened sequence is also constant.
- A301763Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n.
- A301762Number of ways to choose a constant rooted partition of each part in a rooted partition of n.
- A301761Number of ways to choose a rooted partition of each part in a constant rooted partition of n.
- A301760Number of rooted twice-partitions of n where the composite rooted partition is constant.
- A301756Number of ways to choose disjoint strict rooted partitions of each part in a strict rooted partition of n.
- A301754Number of ways to choose a strict rooted partition of each part in a strict rooted partition of n.
- A301753Number of ways to choose a strict rooted partition of each part in a rooted partition of n.
- A301751Number of ways to choose a rooted partition of each part in a strict rooted partition of n.
- A301750Number of rooted twice-partitions of n where the composite rooted partition is strict.
- A301706Number of rooted thrice-partitions of n.
- A301700Number of aperiodic rooted trees with n nodes.
- A301598Number of thrice-factorizations of n.
- A301595Number of thrice-partitions of n.
- A301481Number of unlabeled uniform hypergraphs spanning n vertices.
- A301480Number of rooted twice-partitions of n.
- A301470Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.
- A301469Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.
- A301467Number of enriched r-trees of size n with no empty subtrees.
- A301462Number of enriched r-trees of size n.
- A301422Regular triangle where T(n,k) is the number of r-trees of size n with k leaves.
- A301368Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.
- A301367Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves.
- A301366Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.
- A301365Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.
- A301364Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.
- A301345Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.
- A301344Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.
- A301343Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.
- A301342Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.
- A300913Number of non-isomorphic connected set-systems of weight n.
- A300912Numbers of the form prime(x)*prime(y) where x and y are relatively prime.
- A300866Signed recurrence over binary strict trees: a(n) = 1 - Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).
- A300865Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).
- A300864Signed recurrence over strict trees: a(n) = -1 + Sum_{y1 + ... + yk = n, y1 > ... > yk > 0, k > 1} a(y1) * ... * a(yk).
- A300863Signed recurrence over enriched p-trees: a(n) = (-1)^(n - 1) + Sum_{y1 + ... + yk = n, y1 >= ... >= yk > 0, k > 1} a(y1) * ... * a(yk).
- A300862Solution to 1 = Sum_y Product_{k in y} a(k) for each n > 0, where the sum is over all integer partitions of n with an odd number of parts.
- A300797Number of strict trees of weight 2n + 1 in which all outdegrees and all leaves are odd.
- A300789Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.
- A300788Number of strict integer partitions of n in which the even parts appear as often at even positions as at odd positions.
- A300787Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.
- A300652Number of enriched p-trees of weight 2n + 1 in which all outdegrees and all leaves are odd.
- A300650Number of orderless same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.
- A300649Number of same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.
- A300648Number of orderless same-trees of weight n in which all outdegrees are odd.
- A300647Number of same-trees of weight n in which all outdegrees are odd.
- A300626Number of inequivalent colorings of orderless Mathematica expressions with n positions.
- A300575Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...
- A300574Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).
- A300486Number of relatively prime or monic partitions of n.
- A300443Number of binary enriched p-trees of weight n.
- A300442Number of binary strict trees of weight n.
- A300440Number of odd strict trees of weight n (all outdegrees are odd).
- A300439Number of odd enriched p-trees of weight n (all outdegrees are odd).
- A300436Number of odd p-trees of weight n (all outdegrees are odd).
- A300385In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.
- A300384In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.
- A300383In the ranked poset of integer partitions ordered by refinement, a(n) is the size of the lower ideal generated by the partition with Heinz number n.
- A300355Number of enriched p-trees of weight n with odd leaves.
- A300354Number of enriched p-trees of weight n with distinct leaves.
- A300353Number of strict trees of weight n with odd leaves.
- A300352Number of strict trees of weight n with distinct leaves.
- A300351Triangle whose n-th row lists in order all Heinz numbers of integer partitions of n into odd parts.
- A300335Number of ordered set partitions of {1,...,n} with weakly increasing block-sums.
- A300301Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.
- A300300Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.
- A300273Heinz numbers of collapsible integer partitions.
- A300272Heinz numbers of odd partitions.
- A300271Smallest Heinz number of a partition obtained from y by removing one square from its Young diagram, where y is the integer partition with Heinz number n > 1.
- A300124Number of ways to tile the diagram of an integer partition of n using connected skew partitions.
- A300123Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.
- A300122Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.
- A300121Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.
- A300120Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.
- A300118Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.
- A300063Heinz numbers of integer partitions of odd numbers.
- A300061Heinz numbers of integer partitions of even numbers.
- A300060Number of domino tilings of the diagram of the integer partition with Heinz number n.
- A300056Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.
- A299968Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.
- A299967Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions non-singleton skew-partitions.
- A299966Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions non-singleton skew-partitions.
- A299926Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions skew partitions.
- A299925Number of chains in Young's lattice from () to the partition with Heinz number n.
- A299764Number of special products of factorizations of n into factors > 1.
- A299759Triangle whose n-th row lists in order all FDH numbers of strict integer partitions of n.
- A299758Largest FDH number of a strict integer partition of n.
- A299757Weight of the strict integer partition with FDH number n.
- A299756Triangle whose n-th row is the finite increasing sequence, or set of positive integers, with FDH number n.
- A299755Triangle whose n-th row is the strict integer partition with FDH number n.
- A299729Heinz numbers of non-knapsack partitions.
- A299702Heinz numbers of knapsack partitions.
- A299701Number of distinct subset-sums of the integer partition with Heinz number n.
- A299699Number of rim-hook (or border-strip) tableaux of size n.
- A299471Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.
- A299354Regular triangle where T(n,k) is the number of labeled connected k-uniform hypergraphs spanning n vertices.
- A299353Number of labeled connected uniform hypergraphs spanning n vertices.
- A299203Number of enriched p-trees whose multiset of leaves is the integer partition with Heinz number n.
- A299202Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.
- A299201Number of twice-partitions whose composite is the integer partition with Heinz number n.
- A299200Number of twice-partitions whose domain is the integer partition with Heinz number n.
- A299152Denominators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).
- A299151Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).
- A299150Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).
- A299149Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).
- A299119Positive solution to 2^(n-1) = 1/n * Sum_{d|n} a(d) * a(n/d).
- A299090Number of "digits" in the binary representation of the multiset of prime factors of n.
- A299072Sequence is an irregular triangle read by rows with zeroes removed where T(n,k) is the number of compositions of n whose standard factorization into Lyndon words has k distinct factors.
- A299070Regular triangle T(n,k) is the number of compositions of n whose standard factorization into Lyndon words has k distinct factors.
- A299027Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.
- A299026Number of compositions of n whose standard factorization into Lyndon words has all weakly increasing factors.
- A299024Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.
- A299023Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.
- A298971Number of compositions of n that are proper powers of Lyndon words.
- A298947Number of integer partitions y of n such that exactly one permutation of y is a Lyndon word.
- A298941Number of permutations of the multiset of prime factors of n > 1 that are Lyndon words.
- A298748Heinz numbers of aperiodic (relatively prime multiplicities) integer partitions with relatively prime parts.
- A298540Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of nodes.
- A298539Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of nodes.
- A298538Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of nodes.
- A298537Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.
- A298536Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of leaves.
- A298535Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of leaves.
- A298534Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of leaves.
- A298533Number of unlabeled rooted trees with n vertices such that every branch of the root has the same number of leaves.
- A298479Matula-Goebel numbers of rooted trees in which all positive outdegrees are different.
- A298478Number of rooted trees with n nodes in which all positive outdegrees are different.
- A298426Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.
- A298424Matula-Goebel numbers of rooted trees in which all positive outdegrees are the same.
- A298423Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.
- A298422Number of rooted trees with n nodes in which all positive outdegrees are the same.
- A298363Matula-Goebel numbers of rooted identity trees with thinning limbs.
- A298305Matula-Goebel numbers of rooted trees with strictly thinning limbs.
- A298304Number of rooted trees on n nodes with strictly thinning limbs.
- A298303Matula-Goebel numbers of rooted trees with thinning limbs.
- A298262Number of integer partitions of n using relatively prime non-divisors of n.
- A298207Numbers that are a product of zero, one, or three prime numbers.
- A298205Matula-Goebel numbers of rooted trees in which all outdegrees are either 0, 1, or 3.
- A298204Number of semi-ternary rooted trees with n vertices.
- A298126Matula-Goebel numbers of rooted trees in which all outdegrees are even.
- A298120Matula-Goebel numbers of rooted trees in which all positive outdegrees are odd.
- A298118Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.
- A297791Number of series-reduced leaf-balanced rooted trees with n nodes. Number of orderless same-trees with n nodes and all leaves equal to 1.
- A297571Matula-Goebel numbers of fully unbalanced rooted trees.
- A296978Normal sequences ordered first by length and then lexicographically, where a finite sequence is normal if it spans an initial interval of positive integers.
- A296977Normal Lyndon sequences ordered first by length and then lexicographically, where a finite sequence is normal if it spans an initial interval of positive integers.
- A296976Normal Lyndon sequences ordered first by length and then reverse-lexicographically, where a finite sequence is normal if it spans an initial interval of positive integers.
- A296975Number of aperiodic normal sequences of length n.
- A296774Triangle read by rows in which row n lists the compositions of n ordered first by length and then reverse-lexicographically.
- A296773Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then lexicographically.
- A296772Triangle read by rows in which row n lists the compositions of n ordered first by decreasing length and then reverse-lexicographically.
- A296659Length of the final word in the standard Lyndon word factorization of the first n terms of A000002.
- A296658Length of the standard Lyndon word factorization of the first n terms of A000002.
- A296657Triangle whose n-th row is the concatenated sequence of all binary Lyndon words of length n in lexicographic order.
- A296656Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in reverse-lexicographic order.
- A296561Number of rim-hook (or border-strip) tableaux whose shape is the integer partition with Heinz number n.
- A296560Number of semistandard Young tableaux whose shape is the conjugate of the integer partition with Heinz number n.
- A296373Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k.
- A296372Triangle T(n,k) = number of normal sequences of length n whose factorization into Lyndon words (aperiodic necklaces) is of length k.
- A296371Number of integer partitions of n using Jacobsthal numbers.
- A296302Number of aperiodic compositions of n with relatively prime parts.
- A296188Number of semistandard Young tableaux whose shape is the integer partition with Heinz number n.
- A296150Triangle whose n-th row is the integer partition with Heinz number n.
- A296134Number of twice-factorizations of n of type (R,Q,R).
- A296133Number of twice-factorizations of n of type (Q,R,Q).
- A296132Number of twice-factorizations of n where the first factorization is constant and the latter factorizations are strict, i.e., type (P,R,Q).
- A296131Number of twice-factorizations of n of type (P,Q,R).
- A296122Number of twice-partitions of n with no repeated partitions.
- A296121Number of twice-factorizations of n with no repeated factorizations.
- A296120Number of ways to choose a strict factorization of each factor in a strict factorization of n.
- A296119Number of ways to choose a strict factorization of each factor in a factorization of n.
- A296118Number of ways to choose a factorization of each factor in a strict factorization of n.
- A295935Number of twice-factorizations of n where the latter factorizations are constant, i.e. type (P,P,R).
- A295931Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.
- A295924Number of twice-factorizations of n of type (R,P,R).
- A295923Number of twice-factorizations of n where the first factorization is constant, i.e. type (P,R,P).
- A295920Number of twice-factorizations of n of type (P,R,R).
- A295636Write 2 - Zeta(s) in the form Product_{n > 1}(1 - a(n)/n^s).
- A295635Write 2 - Zeta(s) in the form 1/Product_{n > 1}(1 + a(n)/n^s).
- A295632Write 1/Product_{n > 1}(1 - 1/n^s) in the form Product_{n > 1}(1 + a(n)/n^s).
- A295461Number of unlabeled rooted trees with 2n + 1 nodes in which all outdegrees are even.
- A295281Number of complete strict tree-factorizations of n > 1.
- A295279Number of strict tree-factorizations of n.
- A294859Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in lexicographic order.
- A294788Number of twice-factorizations of type (Q,P,Q) and product n.
- A294787Number of ways to choose a set partition of a factorization of n into distinct factors greater than one.
- A294786Number of ways to choose a set partition of a factorization of n into distinct factors greater than one such that different blocks have different products.
- A294617Number of ways to choose a set partition of a strict integer partition of n.
- A294339Number of ways to write 2^n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.
- A294338Number of ways to write n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.
- A294337Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.
- A294336Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.
- A294150Number of knapsack partitions of n that are also knapsack factorizations.
- A294080Same-tree Moebius function of the multiorder of integer partitions indexed by Heinz numbers.
- A294079Strict Moebius function of the multiorder of integer partitions indexed by Heinz numbers.
- A294068Number of factorizations of n using perfect powers (elements of A001597) other than 1.
- A294019Number of same-trees whose leaves are the parts of the integer partition with Heinz number n.
- A294018Number of strict trees whose leaves are the parts of the integer partition with Heinz number n.
- A293994Number of unlabeled multiset clutters of weight n.
- A293993Number of unlabeled multiset antichains of weight n.
- A293627Number of knapsack factorizations whose factors sum to n.
- A293607Number of unlabeled clutters of weight n.
- A293606Number of unlabeled antichains of weight n.
- A293511Numbers that can be written as a product of distinct squarefree numbers in exactly one way.
- A293510Number of connected minimal covers of n vertices.
- A293243Numbers that cannot be written as a product of distinct squarefree numbers.
- A292886Number of knapsack factorizations of n.
- A292884Number of ways to shuffle together a multiset of compositions to form a composition of n.
- A292505Number of complete orderless tree-factorizations of n >= 2.
- A292504Number of orderless tree-factorizations of n.
- A292444Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.
- A292432Number of normal multisets that cannot be expressed as the multiset-union of a set of sets.
- A292127a(r(n)^k) = 1+k*a(n) where r(n) is the n-th rootless number.
- A292050Matula-Goebel numbers of semi-binary rooted trees.
- A291686Numbers whose prime indices other than 1 are distinct prime numbers.
- A291636Matula-Goebel numbers of lone-child-avoiding rooted trees.
- A291634Number of unlabeled binary rooted trees with n nodes.
- A291443Number of leaf-balanced trees with n nodes.
- A291442Matula-Goebel numbers of leaf-balanced trees.
- A291441Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.
- A290973Write 2x/(1-x) in the form (1-x)^a(1) - 1 + (1-x^2)^a(2) - 1 + (1-x^3)^a(3) - 1 + ...
- A290971Write x/(1-x) in the form a(1)x/(1+a(1)x) + a(2)x^2/(1+a(2)x^2) + a(3)x^3/(1+a(3)x^3) + ...
- A290822Transitive numbers. Matula-Goebel numbers of transitive rooted trees.
- A290760Matula-Goebel numbers of transitive rooted identity trees.
- A290689Number of transitive rooted trees with n nodes.
- A290320Write 1 - t * x/(1-x) as an inverse power product 1/(1+c(1)x) * 1/(1+c(2)x^2) * 1/(1+c(3)x^3) * ... The sequence is a regular triangle where T(n,k) is the coefficient of t^k in c(n).
- A290262Triangle whose rows give the nonzero coefficients of -t^k (k >= 1) in the inverse power product expansion of 1 - t * x/(1-x).
- A290261Write 1 - x/(1-x) as an inverse power product 1/(1+a(1)x) * 1/(1+a(2)x^2) * 1/(1+a(3)x^3) * ...
- A289501Number of enriched p-trees of weight n.
- A289079Number of orderless same-trees of weight n with all leaves equal to 1.
- A289078Number of orderless same-trees of weight n.
- A289023Position in the sequence of rootless numbers (A007916) of the smallest positive integer x such that for some positive integer y we have n = x^y (A052410).
- A288636Height of power-tower factorization of n. Row lengths of A278028.
- A288605Position of first appearance of each integer in A088568 (number of 1's minus number of 2's in first n terms of A000002).
- A286520Number of finite connected sets of pairwise indivisible positive integers greater than one with least common multiple n.
- A286518Number of finite connected sets of positive integers greater than one with least common multiple n.
- A285573Number of finite nonempty sets of pairwise indivisible divisors of n.
- A285572Number of finite sets of pairwise indivisible positive integers with least common multiple n.
- A285175Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.
- A284640Number of positive subset sums of strict integer partitions of n.
- A284639Number of ways to write n>1 as a power of a product.
- A283877Number of non-isomorphic set-systems of weight n.
- A281146Number of same-trees of weight n with all leaves equal to 1.
- A281145Number of same-trees of weight n.
- A281119Number of complete tree-factorizations of n>=2.
- A281118Number of tree-factorizations of n>=2.
- A281116Number of factorization of n into positive integers greater than 1 with no common divisor other than 1.
- A281113Number of twice-factorizations of n.
- A281013Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.
- A280996Prime numbers p whose index pi(p) is a Matula-Goebel number of a generalized Bethe tree
- A280994Triangle read by rows giving Matula-Goebel numbers of planted achiral trees with n nodes.
- A280962Number of integer partitions of even/odd numbers using primes minus one.
- A280954Number of integer partitions of n using predecessors of prime numbers
- A280000Number of free pure symmetric multifunctions in one symbol with n positions.
- A279984Positions of the prime numbers in the sequence of rootless numbers.
- A279969a(1)=1, a(n+1)=2^(prime(a(n))-1).
- A279944Number of positions in the free pure symmetric multifunction in one symbol with j-number n.
- A279863Number of maximal transitive finitary sets with n brackets.
- A279861Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.
- A279791Number of twice-partitions of type (Q,R,Q) and weight n.
- A279790Number of twice-partitions of type (Q,P,Q) and weight n.
- A279789Number of ways to choose a constant partition of each part of a constant partition of n.
- A279788Twice partitioned numbers where the first partition is constant and the latter partitions are strict.
- A279787Twice partitioned numbers where the first partition is constant.
- A279786Twice partitioned numbers where the first partition is strict and the latter partitions are constant.
- A279785Number of ways to choose a strict partition of each part of a strict partition of n.
- A279784Twice partitioned numbers where the latter partitions are constant.
- A279614a(1)=1, a(d(x_1)*..*d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.
- A279375Number of set partitions of strict integer partitions of n that have distinct block-sums.
- A279374Number of ways to choose an odd partition of each part of an odd partition of 2n-1.
- A279065Fermi-Dirac primeth recurrence.
- A277996Number of distinct orderless Mathematica expressions with one atom and n positions.
- A277615a(1)=1, a(c(x_1)^...^c(x_k))=1+a(x_1)+...+a(x_k), where c(n) is the n'th not perfect power.
- A277576Radical paths. a(n) = rad(a(n-1)) where rad(n) is the n'th radical number.
- A277562Perfect towers.
- A277427Concatenated sequence of all prime permutations ordered lexicographically.
- A277098Finitary primes. Primes of finitary index.
- A276687Number of prime plane trees of weight A000040(n).
- A276625Finitary numbers. Matula-Goebel numbers of rooted identity trees.
- A276024Number of positive subset sums of integer partitions of n.
- A275972Number of strict knapsack partitions of n.
- A275870Number of collapsible integer partitions of n.
- A275307Number of labeled spanning blobs on n vertices.
- A275024Total weight of the n-th twice-prime-factored multiset partition.
- A273873Number of strict trees of weight n.
- A273461Number of physically stable n X n placements of water source-blocks in Minecraft.
- A271619Twice partitioned numbers where the first partition is strict.
- A269134Number of combinatory separations of normal multisets of weight n.
- A267597Number of sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of any submultiset of y is distinct.
- A262673Number of pointed trees on normal pointed multisets of weight n.
- A262671Number of pointed multiset partitions of normal pointed multisets of weight n.
- A255397Number of multimin-partitions of normal multisets of weight n.
- A198085Total number of clutters on all subsets of [n].
- A196545Number of weakly ordered plane trees with n leaves.