cp's OEIS Frontend

The Heinz numbers of these partitions are given by A325368.

Also a(n) is the total number of ways to represent n+1 as a centered polygonal number of the form km(m+1)/2+1 for k>1. - Alexander Adamchuk, Apr 26 2007

Number of oblong numbers that divide 2n. - Ray Chandler, Jun 24 2008

The number of divisors d of 2n such that d+1 is also a divisor of 2n, see first formula. - Michel Marcus, Jun 18 2015

From Gus Wiseman, May 03 2019: (Start)

Also the number of integer partitions of n forming a finite arithmetic progression with offset 0, i.e. the differences are all equal (with the last part taken to be 0). The Heinz numbers of these partitions are given by A325327. For example, the a(1) = 1 through a(12) = 3 partitions are (A = 10, B = 11, C = 12):

1 2 3 4 5 6 7 8 9 A B C

21 42 63 4321 84

321 642

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From Gus Wiseman, Sep 13 2022: (Start)

Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:

{{1}} {{2}} {{3}} {{4}}

{{1},{1}} {{1,2}} {{1},{3}}

{{1},{2}} {{2},{2}}

{{1},{1},{1}} {{1},{1,2}}

{{1},{1},{2}}

{{1},{1},{1},{1}}

Intervals are counted by A001227, ranked by A073485.

The initial version is A007294.

The strict version is A327731.

The version for gapless multisets instead of intervals is A356941.

The case of strict partitions is A356957.

Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:

{{1}} {{2}} {{3}} {{4}}

{{1,1}} {{1,1,1}} {{2,2}}

{{1},{2}} {{1},{3}}

{{1},{1,1}} {{1,1,1,1}}

{{2},{1,1}}

{{1},{1,1,1}}

Constant multisets are counted by A000005, ranked by A000961.

The non-strict version is A006171.

The unlabeled version is A089259.

The non-constant block version is A261049.

The version for twice-partitions is A279786, factorizations A296131.

Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:

{{1}} {{2}} {{3}} {{4}}

{{1},{1}} {{1,1,1}} {{1},{3}}

{{1},{2}} {{2},{2}}

{{1},{1},{1}} {{1},{1,1,1}}

{{1},{1},{2}}

{{1},{1},{1},{1}}

The strict version is A327731 (also).

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Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - Emeric Deutsch, Mar 23 2005

Euler transform of the triangular numbers 1,3,6,10,...

Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - Gary W. Adamson, Jun 13 2009

The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - Vaclav Kotesovec, May 12 2015]

Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - Alimzhan Amanov, Jul 13 2021

Also the number of integer partitions of n whose parts cover an initial interval of positive integers with distinct multiplicities. Also the number of integer partitions of n whose multiplicities cover an initial interval of positive integers and are distinct (see A048767 for a bijection). - Gus Wiseman, May 04 2019

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.

Generating function of the "second integrals" of partitions: given a partition (p_1, ..., p_s) written in weakly decreasing order, write the sequence B = (b_1, b_2, ..., b_s) = (p_1, p_1 + p_2, ..., p_1 + ... + p_s). The sequence gives the coefficients of the generating function summing q^(b_1 + ... + b_s) over all partitions of all nonnegative integers. - William J. Keith, Apr 23 2022

From Gus Wiseman, Jan 17 2023: (Start)

Equivalently, a(n) is the number of multisets (weakly increasing sequences of positive integers) with weighted sum n. For example, the Heinz numbers of the a(0) = 1 through a(15) = 7 multisets are:

1 2 3 4 7 6 8 10 15 12 16 18 20 26 24 28

5 11 9 17 19 14 21 22 27 41 30 32

13 23 29 31 33 55 39 34

25 35 37 43 45

49 77 47

65

121

These multisets are counted by A264034. The reverse version is A007294. The zero-based version is A359678.

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