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# User:Bernard Schott

Engineer, retired from Shell.

Number theory, geometry, group theory and ring theory, curves, olympiad problems, recreational mathematics, limits in analysis, history of mathematics, Diophantine equations, ...
And also interested with beautiful mathematics, surprising theorems, nice proofs, pretty formulas, tricky equations, ...
Author of "Les nombres brésiliens" in Quadrature, no. 76, avril-juin 2010 : A125134.

== Sampling of some contributions: new sequences or comments or formulas [in progress]

## Some families of sequences

``` 1.2 Number of rings with n elements
```
• There exist rings with or without 1 (the multiplicative identity element), and also there exist rings that are commutative or not commutative. There were in OEIS six sequences that gave the number of rings with n elements. I have proposed the three last not existing sequences, those for rings without 1, with examples and explanations: A342375, A342376, A342377.
• Also, the 9 possible cases of rings appear now in the table in Crossrefs section of A027623.

``` 1.3 n-phile and n-phobe numbers
```
• The idea for these sequences and the words 'n-phile' and 'n-phobe' come from the generalization of the problem A496 proposed on French site Diophante (see link in A019532).
• For n >1, an integer m is called n-phile if there exist n positive integers b_1 < b_2 < ... < b_j < ... < b_n such that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n.
• An integer that is not n-phile is called n-phobe.

n-phobe numbers: A019532 (n=3), A348519 (n=4), A348520 (n=5).

n-phile numbers: A160811 \ {5} (n=3), A348517 (n=4), A348518 (n=5).

• Properties:

The number of n-phobe numbers is always finite = A349189(n).

The smallest n-phobe number is always 1 and the largest one is A349188(n).

The smallest n-phile number is 2^n - 1, and there are infinitely many n-phile numbers.

``` 1.4 Equation (x+y) + (x-y) + (x*y) + (x/y) = z with y | x
```
• Generalization of a problem proposed by Yakov Perelman in his book: Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132
• Consider the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = z, when x and y are both positive integers with y | x, or x = K*y.
• Then, there is a solution (x, y) iff z is a term of A013929: numbers that are not squarefree.
• In this case, with x = K*y, then z = S(K*y, y) = K*(y+1)^2; see the table T(n, k) = n*(k+1)^2 in A351381; here (K->n) and (y->k) to get table T(n, k) in OEIS; example: S(12, 4) = T(3, 4) = 75 = a(28).
• The number of solutions (x, y) for S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(n) is A353282(n).
• The smallest nonsquarefree number m such that equation S(x,y) = m has exactly n solutions, for n >= 0, is A130279(n+1).
• Integers k for which number of solutions to the equation S(x,y) = k sets a new record are in A046952 = squares of highly composite numbers.
``` 1.5 Equation k * M = 1M1
```

The reason for this family of sequences comes from the 21-digit integer 112359550561797732809 found in Penguin dictionary with this property: "The smallest number which, when 1 is placed at both ends, the number is multiplied by 99". Natural question was: are there other integers sharing this property?

• A329914: Terms of this sequence are the other numbers k that have the same property than 99 and satisfy: k * M = 1M1, that is concatenation of 1, M and 1. It is surprising that this sequence is full with 15 terms.
• A329915: In fact, for each such k in A329914, there exist an infinite set of integers {M_k} such that k * M_k = 1M_k1 and the smallest terms M_k for each corresponding k are listed in this sequence that is also full with 15 terms.

Some sequences corresponding to miscellaneous values of k:

• A095372 \ {1} = Numbers M such that 21 * M = 1M1, except for a(0) = 1.
• A331630 = Numbers M such that 23 * M = 1M1.
• A351237 = Numbers M such that 83 * M = 1M1.
• A351238 = Numbers M such that 87 * M = 1M1.
• A351239 = Numbers M such that 101 * M = 1M1.

Two last sequences:

• A116436 = Numbers m which when sandwiched between two 1's give a multiple of m.
• A351320 = a(n) is the unique integer k such that k * A116436(n) = 1||A116436(n)||1 where "||" stands for concatenation. Except for a(1) = 111, which is unique, all terms appear infinitely many times and belong to this set of fifteen integers: {21, 23, 27, 29, 33, 39, 57, 59, 69, 71, 83, 87, 99, 101, 107}; see A329914.
``` 1.6 Euclidean division and geometric progression
```

Let m = d*q + r be the Euclidean division of m by d; in addition, for which integers m, triples (r, q, d) or (r, d, q) or (q, r, d) form a geometric progression in this order? (Generalization of Project Euler, Problem 141).

• A334185: Geometric progression is (r, q, d) with a common integer ratio b > 1.
• A334186: Geometric progression is (r, q, d) with a noninteger common ratio b > 1.

Note: in these three previous cases, m is a term since m = d*q + r with r < q < d in geometric progression; but also, with d' = q and q' = d, we have the other Euclidean division m = d'*q' + r with r < d' < q'. So, for all terms m of these three sequences, there exist these two geometric progressions (r, q, d) and (r, d', q') with samme common ratio b..

• A001093 \ {0, 1, 2}: Subsequence of A334185 since n^3 + 1 = n^2 * n + 1 (r=1, q=n, d=n^2, so b = n).
• A002378: For oblong numbers, there exists always a geometric progression (q, r, d) with a common integer ratio b > 1 since k(k+1) = k^2 *1 + k with q = 1, r = k and d = k^2, so b = k.
• A335064: Subsequence of oblong where there exists also an other geometric progression (q, r, d) with a noninteger common ratio b > 1. For these terms m of A335064, m = k*(k+1) with k in A024619.
• A335065: Integers m such that there exists a geometric progression (r, q, d) or (q, r, d) or (r, d, q); so, for these integers m, at least one of these 3 geometric progressions is present.
• A335272: Integers m such that there exist three Euclidean divisions of m by d, d', and d", m = d*q + r = d'*q' + r' = d"*q" + r", such that (r, q, d), (r', d', q'), and (q", r", d") are three geometric progressions; so, for these integers, these 3 geometric progressions are present.
``` 1.7 When n\$ / m! is a square
```

These sequences result from a theory coming from the generalization of a problem proposed in 1996 during two distinct mathematical competitions (Tournament of Towns and Moscow Mathematical Olympiad); the problem asked was:

For which m, 1 <= m <= 100, 100\$ / m! is a perfect square where 100\$ = 1! * 2! * 3! * ... * 100! ?

Many proofs are in the article of Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352 (see link).

• A000178 = n\$ = 1!*2!*...*n! = superfactorial: product of first n factorials is never a square.
• A348692 = Triangle whose n-th row lists the integers m such that n\$ / m! is a square; if there is no such m, then n-th row = 0.
• A349079 = Numbers k such that there exists m, 1 <= m <= k with the property that k\$ / m! is a square.
• A349080 = Numbers k for which there exists only one integer m with 1 <= m <= k such that k\$ / m! is a square: this sequence is the union of {1} and of three infinite and disjoint subsequences:

--> RECYCLED = Numbers k divisible by 4 but not of the form 8q^2 or 8q(q+1) = {4, 12, 20, 24, 28, ...}. For these numbers, the corresponding unique m = k/2 (see example for k = 4). This sequence has been recycled because 4 times another sequence, this recycling is a sad decision for the theory about this nice olympiad problem. That makes a hole in this puzzle of ten constructive sequences; however, this sequence is accessible here in https://oeis.org/history/view?seq=A349494&v=26

--> RECYCLED = Even numbers k not divisible by 4 and of the form k = 2*A055792 = 2*q^2, with q>1 in A001541 = {18, 578, ...}. For these numbers, the corresponding unique m = k/2 – 2 = q^2-2 (see example for k = 18). This sequence has been recycled because 2 times another sequence, same remark as before; however, this sequence is accessible here in https://oeis.org/history/view?seq=A349495&v=26

--> A349496 = Numbers k for which there exists only one integer m with here m = k/2 + 1 such that k\$ / m! is a square, <=> Numbers of the form 4*t^2-2 (A060626) when t >= 1 is an integer that is not a term in A001542 (see example for k = 34).

• A349081 = Numbers k for which there exist two integers m with 1 <= m_1 < m_2 <= k such that k\$ / m! is a square: this sequence is the union of three infinite and disjoint subsequences:

--> A139098 = Numbers k = 8t^2 > 0; for these numbers, m_1 = k/2 – 1 = 4t^2-1 < m_2 = k/2 = 4t^2 (see example for k = 8).

--> A035008 = Numbers k = 8t*(t+1); for these numbers, m_1 = k/2 = 4t(t+1) < m_2 = k/2 + 1 = (2t+1)^2 (see example for k = 16).

--> A349766 = Even numbers K of the form 2t^2-4, t>1 in A001541; for these numbers, m_1 = k/2 + 1 = t^2 – 1 < m_2 = k/2 + 2 = t^2 (see example for k = 14).

## Group theory

``` 2.1 Simple groups
```
• A119648 = Orders for which there is more than one simple group: comment about a(1) = 20160 = 8!/2 and description of these two simple groups PSL_4(2) ~ Alt(8) and PSL_3(4).
• A137863 = Orders of simple groups which are non-cyclic and non-alternating: inserted a(16) = 20160 = 8!/2 with explanation: PSL_3(4) is not isomorphic to Alt(8) + detailed 2 examples: a(1) and a(12).
• A335419 = Integers m such that every group of order m is not simple.
``` 2.2 Number and order of some groups
```
• A003277 = Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k. Proposed comment with link: squarefree terms of A056867 (nilpotent numbers).
• A024619 = Numbers that are not powers of primes p^k (k>=0). Put a theorem from Donald McCarthy with link: (if d is a term) there exists a finite group whose order is divisible by d but which contains no subgroup of order d.
• A030078 = Cube of primes. Proposed comment about the 5 groups of order p^3.
• A054397 = Numbers m such that there are precisely 5 groups of order m. Proposed comment about m = 2*p^2 (A079704 + A143928) and m = p^3 (A030078).
• A147848 = Number (up to isomorphism) of groups of order 2n that have Z/nZ as a subgroup. Proposed example with a(4) = 4 among the 5 groups of order 2*4 = 8.
• A178498 = Number of Frobenius groups of order n. Detailed comment with the two infinite families of Frobenius groups.
• A220211 = The order of the one-dimensional affine group in the finite fields F_q with q >= 3.
• A221048 = The odd semiprime numbers (A046315) which are orders of a non-Abelian group. Proposed comment with the semidirect products of Z/qZ and Z/pZ.
• A350152 = Abelian orders m for which there exist at least 2 groups with order m.
• A350586 = Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.
```2.3 from Des MacHale works
```
• A051532 = Abelian numbers: see similar comment with "squarefree terms" in A003277.
• A056867 = Nilpotent numbers: m such that every group of order m is nilpotent. Proposed an iff with Des MacHale.
• A056868 = Numbers that are not nilpotent numbers; detailed examples for orders 6 and 10.
• A340511 = Numbers k such that there exists a group of order k which has no subgroup of order d, for some divisor d of k; proposed comment + more terms a(35)-a(53) and creation of A341048.
• A341048 = Numbers m such that there is a group of order m that is not supersolvable (NSS) but “converse Lagrange theorem” (CLT).
• A341823 = Number of finite groups G with |Aut(G)| = 2^n. Detailed example a(3) = 7.
• A341824 = Number of groups of order 2^n which occur as Aut(G) for some finite group G. Detailed example a(3) = 3.
• A341825 = Number of finite groups G with |Aut(G)| = n. Detailed example a(6) = 6.
• A349930 = a(n) is the number of groups of order A340511(n) which have no subgroup of order d, for some divisor d of A340511(n).
``` 2.4 Groups of order p^2*q, p != q primes
```
• A000001: Number of groups of order n. Proposed in Formula section a table for the number of groups with order p^2*q.
• A054753: Numbers of the form p^2 * q. Put in Crossrefs a table giving for each subsequence the corresponding number of groups of order p^2*q.
• A079704: a(n) = 2*prime(n)^2. Comment: for these numbers m, there are precisely 5 groups of order m with description of these groups (p = 2 and odd p are distinct cases).
• A143928: Numbers of the form 2*p^2, for p an odd prime. Comment: for these numbers m, there are precisely 5 groups of order m that are described.
• A349495: Numbers p^2*q, p<q primes such that p divides q-1 and p^2 does not divide q-1, with (p,q) <> (2,3). Comment: for these terms m, there are precisely 4 groups of order m, described, so this is a subsequence of A054396.
• A350115: Numbers of the form m = p^2*q, p < q primes such that p^2 divides q-1: for these terms m, there are precisely 5 groups of order m, described, so this is a subsequence of A054397.
• A350245: Numbers of the form m = p^2*q, p > q odd primes such that q divides p+1 : for these terms m, there are precisely 3 groups of order m, described, so this is a subsequence of A055561
• A350332: Numbers of the form m = p^2*q, p < q odd primes such that p does not divide q-1 : for these terms m, there are precisely 2 groups of order m, described, so this is a subsequence of A054395.
• A350638: Numbers of the form m = p^2*q, with odd primes p > q, such that q divides p-1 : for these terms m there are precisely (q+9)/2 groups of order m, described.
• A350421: Numbers of the form m = p^2*q, p > q odd primes such that q does not divide p-1, and q does not divide p+1, for these terms m, there are precisely 2 groups of order m, descrobed, so this is a subsequence of A054395.
• A350422: Numbers of the form m = p^2*q for which there exist exactly 2 groups of order m, described. Equals A350332 Union A350421 = terms with p < q in A350332 and terms with p > q in A350421, with p, q odd primes.
``` 2.5 Linear group
```
• A000056 = Order of the group SL(2, Z_n), comment: SL(2,Z_2) is isomorphic to the symmetric S_3.
• A334884 = Order of the non-isomorphic groups PSL(m,q) [or PSL_m(q)] in increasing order as q runs through the prime powers. Terms are repeated only when two groups with the same order are non-isomorphic; example: a(18) = a(19) = 20160 for PSL(4,2) == A_8 and for PSL(3,4) non-isomorphic to A_8, where == stands for "isomorphic to".
• A335384 = Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.
``` 2.6 Symmetric group
```
• A051625 = Number of “labeled” cyclic subgroups of symmetric S_n; examples for S_3 et S_4.
• A088436 = Number of permutations in the symmetric group S_n that have exactly one transposition in their cycle decomposition; examples for S_4 and S_5.

## Some particular numbers

``` 3.1 Zuckerman numbers: A007602
```

3.1.1 Some sequences about the quotients

• A288069: Quotients obtained when the Zuckerman numbers are divided by the product of their digits.
• A342593: Numbers that are not the quotient of a Zuckerman numbers divided by the product of its digits [similar for Niven numbers is A003635]. As all numbers ending with 0 are terms ...:
• A342941: Numbers not ending with 0, that are not the quotient of a Zuckerman number divided by the product of its digits.
• A343681: Zuckerman numbers which when divided by product of their digits, give a quotient which is also a Zuckerman number.
• A343744: Zuckerman numbers which divided by the product of their digits give integers which are also divisible by the product of their digits, and so on, until result is 1 (fini + full) [similar for Niven numbers is A114440].

3.1.2 Zuckerman numbers: divisors and prime factors

• A335037: a(n) is the number of divisors of n that are Zuckerman numbers.
• A335038: a(n) is the smallest number m with exactly n divisors that are Zuckerman numbers , or -1 if there is no such m.
• A340638: Integer whose number of divisors that are Zuckerman numbers sets a new record [similar for Niven numbers is A340367].
• A337941: Numbers whose divisors are all Zuckerman numbers. (This sequence is infinite if and only if there are infinitely many repunit primes).
• A359961: Smallest Zuckerman number with exactly n distinct prime factors [similar for Niven numbers is A359960].
``` 3.2 Niven numbers: A005349
```

3.2.1 Some sequences about the quotients

• A334416 + A334417 : Numbers m such that (m / sum of digits of m) is a palindrome + corresponding quotients.
• A340637: Integers whose number of divisors that are Niven numbers sets a new record [similar for Zuckerman numbers is A340638].
• A342650: Niven numbers that are divisible by their nonzero digits.
• A342262: Niven numbers that are divisible by the product of their nonzero digits.
• A358067: a(n) is the smallest m such that A144261(m) = n <=> a(n) is the smallest element m of the set of the integers k that satisfy {A144261(k) = n and n * k is a Niven number}.

1.2.2 Niven numbers with n digits

• A348318: Number of n-digit Niven (or Harshad) numbers not containing the digit 0 (Olympiad).
• A348150: a(n) is the smallest Niven (or Harshad) number with exactly n digits and not containing the digit 0.
• A348317: a(n) = A348150 (n) - R_n where R_n is the repunit with n times digit 1 <=> a(n) = the gap between the smallest n-digit number not containing the digit 0 and the smallest n-digit Niven number not containing the digit 0.
• A348316: a(n) is the largest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

3.2.3 Primitive Niven numbers

• A356349: Primitive Niven numbers: terms of A005349 that are not ten times another term of A005349 [with Rémy Sigrist].
• A358255: Primitive Niven numbers ending with zero.
• A358256: a(n) is the smallest primitive Niven number ending with n zeros.

3.2.4 Niven numbers: divisors and prime factors

• A340637: Integers whose number of divisors that are Niven numbers sets a new record [similar for Zuckerman numbers is A340638].
• A359960: Smallest Niven number with exactly n distinct prime factors [similar for Zuckerman numbers is A359961].
• A360011: Integers k such that the product of the first k primes is a Niven number (comment).
``` 3.3 Zuckerman & Niven numbers:
```
• A343680: Niven numbers which when divided by the sum of their digits, give a quotient which is a Zuckerman number.
• A343682: Zuckerman numbers which when divided by the product of their digits, give a quotient which is a Niven number.
``` 3.4 Zuckerman & Smith numbers:
```
• A351618: Numbers that are both Zuckerman numbers and Smith numbers.

## Miscellaneous isolated sequences

``` 4.1 Easy definitions but with interesting or surprising results
```
• A332785 = Nonsquarefree numbers that are not squareful.

Why? Sometimes, nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false, this sequence lists these counterexamples.

• A330616 = Palindromes that are the product of 2 non-palindromic numbers.

Why? Since a little contradiction: 272 =16*17 is a term but 282=6*47 is not a term.

• A339676 = Nonpalindromic numbers that are products of repunits

Why? Since a little contradiction: the first term is A308365 (19) = 161051. According to conjecture of G.J Simmons , these perfect powers are terms: {11^k, k>=4}, {111^k, k>=4}, {1111^k, k>=3}, {11111^k, k>=3}, ...

• A307019 = Squares which can be expressed as the product of a number and its reversal in exactly three different ways.

Why? All terms end with an even number of zeros, and we do not know if a square number can be expressed as the product of a number and its reversal in exactly four different ways.

Example: 6350400 = (2520)^2 = 25200 * 252 = 14400 * 441 = 44100 * 144.

• A342994 = 660, 660660, 660660660, 660660660660, ...

Why? These numbers are exactly integers with trailing zeros whose square can be expressed as the product of a number ending with 0 and its reversal, and also as the product of a number and its reversal, but this time without trailing zero.

Example: a(1) = 435600 = 660^2 = 6600 * 66 = 528 * 825.

• A346274 = Number of n-digit primes with digital product = 7.

Why? a(n) = n iff n = 1 (for 7) or n = 2 (for 17, 71); a(n) < n for n >= 3, since in this case, there is always at least one composite number among the n-digit numbers with digital product = 7 (31st IMO in 1990).

• A342049 = Primes formed by the concatenation of exactly two consecutive composite numbers.

Why? Mix of primes vs composites. The first composite always ends with 0, 2, 6, 8 while the second one ends respectively with 1, 3, 7, 9.

• A358270 = Integers whose sum of digits is even and that have an even number of even digits.

Why? The conditions separately are A054683 for even sum of digits, and A356929 for even number of even digits, so that this sequence is their intersection, while the opposite conditions, an odd sum of digits, and an odd number of odd digits, are the same and are in A054684.

• A348832 = Positive numbers whose square starts and ends with exactly 444.

Why? When a square ends in exactly three identical digits, these digits are necessarily 444, and a(1) = 666462 since it is the smallest term that satisfies 666462^2 = 444171597444.

• A309101 = Primes whose decimal representation can be written as a sequence of primes separated by single zeros.

Why? Because nice term 2030507011013017019023029031037041043047053059061067071073079083089097 consists of the sequence of primes less than 100 separated by zeros.

• A308335 = Palindromic primes such that sum of digits = number of digits.

Why? Except for a(1) = 11, the terms of this sequence must have an odd number of digits that is congruent to 1 or 5 mod 6 (A007310), and moreover, the middle digit of a(n) is odd.

• A342304 = k-digit positive numbers exactly one of whose substrings is divisible by k.

Why? Some properties: any number with two or more 0 digits is not a term; the 2-digit terms are odd; the 5-digit terms are numbers starting with 5, and with no other digits 5 or 0. There are no 10-digit terms.

Why? Every odd number >= 3 is trapezoidal and powers of 2 are not trapezoidal, but we do not know if there exists a finite number of even terms (largest found by Bert Dobbelaere is 48).

• A343591 = Smoothy undulating alternating primes, or primes of the form ababab... with a <> b (A032758) and in which parity of the digits a and b alternates (A030144).

Why? Some properties: every term has two digits or an odd number of digits; in this last case, there exist only 12 possibilities with a odd <> 5 and b even <> 0 to get such primes: 1(21), 1(41), 1(61), 1(81), 3(23), 3(83), 7(27), 7(47), 7(87), 9(29), 9(49), 9(89); all terms with an odd number of digits are palindromic (A059758); only 2 and the nine 2-digit terms start with an even digit. Charles W. Trigg was the first to use the word 'smoothly' for these integers.

``` 4.2 Sequences coming from historical theorems or conjectures or equations or theories
```
• A333635: Numbers m such that m^2 + 1 has at most 2 prime factors <=> m^2 + 1 is prime or semiprime.

Why? Henryk Iwaniec proved in 1978 that this sequence is infinite. By contrast, it is not known whether there are infinitely many primes of the form m^2 + 1 (or infinitely many semiprimes of that form).

• A327802: Number of primes p such that n < p < (9/8) * n.

Why? In 1932, Robert Breusch proved that for n > 47 there is at least one prime p between n and (9/8)*n. This was an improvement of Bertrand's postulate also called Chebyshev's theorem: if n > 1, there is always at least one prime p such that n < p < 2*n.

• A333846: Numbers k such that the number of primes between k^2 and (k+1)^2 increases to a new record.

Why? Legendre's conjecture (still open) states that for k > 0 there is always a prime between k^2 and (k+1)^2, here are listed records for k.

• A339465: Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530

Why? Paul Erdős asked if there are infinitely many primes p such that (p-1)/ A006530(p-1) = 2^k or = 2^q*3^r; A074781 lists the primes corresponding to ratio = 2^k, while this other sequence lists primes corresponding to ratio = 2^q * 3^r. The answers to Erdős questions are not known.

• A346692: a(n) = phi(n) - phi(n-phi(n)), a(1) = 1.

Why? Paul Erdős conjectured that a(n) > 0 on a set of asymptotic density 1, then Luca and Pomerance proved this conjecture.

• A340461: a(n) = 2*sigma(phi(n)) - n.

Why? In 1964, A. Mąkowski and Andrzej Schinzel conjectured that sigma(phi(n))/n >= 1/2 for all n (see links Mąkowski & Schinzel and Graeme L. Cohen), and this conjecture is equivalent to a(n) >= 0. K. Kuhn checked that this inequality holds for all positive integers n having at most six prime factors.

• A220211: The order of the one-dimensional affine group in the finite fields F_q with q >= 3

Why? This family of Frobenius groups belonging to A178498 was not described in OEIS.

• A336819: Odd values of D > 0 for which the generalized Ramanujan-Nagell equation x^2 + D = 2^m has two or more solutions in the positive integers.

Why? Generalization of case D = 7 that corresponds to well known Ramanujan-Nagell equation x^2 + 7 = 2^m with its 5 solutions. If D odd <> 7, Roger Apéry proved in 1960 that the equation x^2 + D = 2^m has at most 2 solutions.

## Nice curves

Some curves with area, length, asymptotic points, maximum curvature, Hausdorff dimension, ...

In fact, each sequence gives the decimal expansion of the corresponding constants.

For many curves, there are links to Mathcurves of Robert Ferréol, or Eric Weisstein's World of Mathematics, or Wikipedia.

Cardioid: A197723 = (3*Pi/2) * a^2 is the area of the cardioid whose polar equation is r = a*(1+cos(t)).

Cissoid of Diocles: A177870 = (3*Pi/4) * a^2 is the area between the cissoid of Diocles and its asymptote when polar equation of cissoid is r = a*sin^2(t)/cos(t).

Cornu spiral: A217481 = If m = (1/2) * sqrt(Pi/2), then the coordinates of the two asymptotic points of the Cornu spiral (also called clothoide) and whose Cartesian parametrization is: x = a * Integral_{0..t} cos(u^2) du and y = a * Integral_{0..t} sin(u^2) du are (a*m, a*m) and (-a*m, -a*m).

Deltoid: A019692 = (2*Pi) * a^2 is the area of the deltoid whose Cartesian parametrization is: (x = a * ((2*cos(t) + cos(2*t)), y = a * ((2*sin(t) - sin(2*t))).

"Double egg": A336266 = (3*Pi/16) * a^2 is the area of one egg of the "double egg" whose polar equation is r(t) = a * cos(t)^2 and a Cartesian equation is (x^2+y^2)^3 = a^2*x^4.

Exponential: A212886 = 2*sqrt(3)/9 is the maximum curvature of function x -> exp(x) while A104956 = 3*sqrt(3)/2. is the corresponding minimum radius of curvature. This maximum curvature occurs at the point M with coordinates [x_M = -(A016655)/10 = -log(2)/2; y_M = A010503 = sqrt(2)/2].

Folium of Descartes: A353049 = (8*sqrt(2)/3)) * (1/a) is the maximum curvature of the Folium of Descartes x^3 + y^3 – 3*a*x*y = 0, occurring at the point M of coordinates (3a/2, 3a/2).

Kepler egg: A336308 = (5*Pi/32) * a^2 is the area of a simple folium also called ovoid, whose polar equation is r = a*cos^3(t) and Cartesian equation is (x^2+y^2)^2 = a * x^3.

Nephroid: A122952 = 3*Pi is the area of the nephroid whose Cartesian parametrization is: x = (1/2) * (3*cos(t) - cos(3t)) and y = (1/2) * (3*sin(t) - sin(3t)).

Newton strophoid: A180434 = (2-Pi/2) * a^2 is the area of the loop of the (also called) right strophoid whose polar equation is r = a*cos(2*t)/cos(t).

Tame twindragon curve: A327620 = 2 * log_2((1+sqrt(78)/9)^(1/3) + (1-sqrt(78)/9)^(1/3)). There exist exactly four regular 2-reptiles in the plane that have fractal boundaries. Only the decimal expansion of the Hausdorff dimension of this dragon curve's boundary was not in OEIS.

Trisectrix of Maclaurin: A010482 = sqrt(27) * a^2 is the area of the loop of this trisectrix and also the area between the curve and its asymptote, when r = 2*a * sin(3t)/sin(2t) is its polar equation.

## Series and constants

1. With 1/n, H(n) and +/-

A347145 = Sum_{n>=1} 1/(n*H(n)^2) (as H(n) ~ log(n), compare with A115563)

A196521 = Sum_{n>=1} (-1)^(n*(n-1)/2) / n = Pi/4 – log(2)/2 (compare with A231902) [formula]

A231902 = Sum_{n>=1} (-1)^(n*(n+1)/2) / n [formula] = Pi/4 + log(2)/2 (compare with A196521) [formula]

A339799 = Sum_{m>=1} (-1)^floor(sqrt(m)) / m = - (1/1+1/2+1/3) + (1/4+1/5+1/6+1/7+1/8) - (1/9+...+1/15) + ...

A353874 = (1/1) - (1/2+1/3) + (1/4+1/5+1/6) - (1/7+1/8+1/9+1/10) + (1/11+1/12+1/13+1/14+1/15) - ... n terms in the n-th group

2. With log(n)

2.1 Bertrand series:

Theorem: Bertrand series Sum_{n>1} 1/(n^q*log(n)^r) is convergent iff (q > 1) or (q=1 and r>1).

-> for (q=1 and r>1): with r = 2, 3, 4, 5 see respectively A115563, A145419, A145420, A145421

-> for (q > 1): for q = 2 with r = -2, -1, 0, 1, 2 see respectively A201994, A073002, A013661, A168218, A349522

2.2 With log and sqrt, n! but without trigonometry

A308915:Sum_{n>=1} 1/(log(n)^log(n))

A336284: Sum_{n>=2} n^(log(n))/log(n)^n

A336741: Sum_{n>=2} 1/log(n)^sqrt(n)

A336730: Sum_{n>=1} log(n)^n / n!

A336987: Sum_{n>=2} sqrt(n)^log(n)/log(n)^sqrt(n)

A351687: Sum_{n>=2} (-1)^n/log(n!)

2.3 With log and trigonometry

A336405: Sum_{n>=1} log(n*sin(1/n)) [negated]

A336603: Sum_{n>=1} log(cos(1/n)) [negated]

A342647: Sum_{n>=1} log(cos(1/n)) * log(sin(1/n))

A353781: Sum_{n>=0} log(cos(1/2^n)) = log(sin(2)/2) [negated]

3. With trigonometry without log

A019987: (Sum_{n=1..90} 2*n*sin(2*n)) / 90, with n in degrees = tan(89 degrees) (link USAMO 1996).

A121225: Sum_{n>=1} cos(n)/n = -log(2-2*cos(1))/2 (Fresnel series)

A096444: Sum_{n>=1} sin(n)/n= (Pi-1)/2 (Fresnel series)

A342680: Sum_{n>=1} sin(sin(n)/n)

A343469: Sum_{n>=1} (-1)^(n-1)/(n*arctan(n))

A343470: Sum_{n>=1} ((-1)^(n-1))*arctan(n)/n

A350885: Sum_{n>=0} (1/2^n) * (tan(1/2^n)) = 1 – 2*cotan(2)

A351738: Sum_{n>0} sin(sqrt(n)) / n

4. With digits

A016627: Sum_{k>=1} A000120(k) / (k*(k+1)) = log(4) (Putnam1981) [base 2]

A334388: Sum_{k>=1} sod(k) / (k*(k+1)) where sod(k) = A007953(k) is the sum of digits of k = (10/9) * log(10) [base 10]

A308314: Sum_{k>=1} (1/A055642(k)^A055642(k)) = (9/10) * Sum_{m>=1} (10/m)^m.

5. With Euler totient phi

Sum_{n>=1} 1/phi(n)^k is convergent iff k > 1: A109695 (k=2), A335818 (k=3) [put this rule]

6. With factorial = Sum_{n>=0} 1/(q*n)!

Formula : for q integer >= 1, Sum_{n>=0} 1/(q*n)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity. For q = 3, 4, 5, 6 see respectively A143819, A332890, A269296, A332892.

7. With Zeta

A338106: Sum_{m>1, n>1} 1/(m^2*n^2-1) = Sum_{k>0} (zeta(2*k) - 1)^2

A338107: Sum_{m>1, n>1} 1/(m^2*n^2+1) = Sum_{k>0} (-1)^(k-1) * (zeta(2*k) - 1)^2

A333972: Sum_{m>0, q>0, m | q} 1/(m^2*q^2) = Pi^6/540 = zeta(2) * zeta(4)

A352527: Sum_(n>=1) (-1)^n * zeta(2n)/(2n) (negated) = log(Pi/sinh(Pi)) / 2

A352619: Sum_{n>=1} (-1)^(n+1) * zeta(2n+1)/(2n+1) = gamma + arg(i!) [question = closed formula?]

8) With primes

A306759: Sum of of reciprocals of Brazilian primes, also called the Brazilian primes constant = Sum_{n>=1} 1/A085104(n)

A338475: Sum of reciprocals of the smallest primes > 2^k for k >= 0

9. Original from curves or equations

A338670: Sum of the negative and positive local extreme values of the sinc function for x > 0 (negated): sinc -> sin(x)/x; while this series is not absolutely convergent, just as (C_1)/2 diverges where C_1 is the corresponding du Bois-Reymond constant.

A354014: Sum_{n>0} u(n) where u(n) is the unique positive solution to the equation Integral_{u(n)..1} e^t/t dt = n.