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# User:Gus Wiseman

From OeisWiki

I am formerly a student of pure mathematics at University of California, Davis. The mainstream applications of my thesis research lie primarily in the field of enumerative graph theory. Graph theory is basically about connected structures, such as trees, cycles, maps, circuits, orientations, circulations, contractions, categories, etc, and this branch of enumerative combinatorics could be described as an analytic and synthetic study of the "chemistry" of the combinatorial (discrete, concrete) connected forms that model novel implementations of mathematical technology.

## 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

- 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?
- 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).
- 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.
- 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?
- 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?
- 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?

## Original Sequences (1550)

- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 series-reduced 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.