cp's OEIS Frontend

Any subsequent terms are > 10^10. - Lucas A. Brown, Jan 06 2023

The Heinz numbers of these partitions are given by A325368.

Rediscovered by the HR automatic theory formation program.

a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001

We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009

Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]

Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015

The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020

The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

Also number of decreasing integer sequences l(1) >= l(2) >= l(3) >= .. 0 such that sum('i*l(i)','i'=1..infinity)=n.

a(n) is also the number of partitions of n such that #{parts equal to i} >= #{parts equal to j} if i <= j.

Also the number of partitions of n (necessarily into distinct parts) where the part sizes are monotonically decreasing (including the last part, which is the difference between the last part and a "part" of size 0). These partitions are the conjugates of the partitions with number of parts of size i increasing. - Franklin T. Adams-Watters, Apr 08 2008

Also partitions with condition as in A179255, and additionally, if more than one part, first difference >= first part: for example, a(10)=7 as there are 7 such partitions of 10: 1+2+3+4 = 1+2+7 = 1+3+6 = 1+9 = 2+8 = 3+7 = 10. - Joerg Arndt, Mar 22 2011

Number of members of A181818 with a bigomega value of n (cf. A001222). - Matthew Vandermast, May 19 2012

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

For n > 0, a(n) is one fewer than the number of maximal runs of the n-th composition in standard-order.

From Gus Wiseman, May 03 2019: (Start)

a(n) is the number of integer partitions of n with equal differences. The Heinz numbers of these partitions are given by A325328. For example, the a(1) = 1 through a(9) = 9 partitions are:

1 2 3 4 5 6 7 8 9

11 21 22 32 33 43 44 54

111 31 41 42 52 53 63

1111 11111 51 61 62 72

222 1111111 71 81

321 2222 333

111111 11111111 432

531

111111111

(End)

From Petros Hadjicostas, Sep 29 2019: (Start)

We show how Leroy Quet's g.f. Sum_{n >= 0} a(n)*x^n = 1/(1-x) + Sum_{k >= 2} x^k/(1-x^(k*(k-1)/2))/(1-x^k) in the Formula section below can be derived from Graeme McRae's g.f. for A049982 (see one of the links below).

Let b(n) = A049982(n) for n >= 1. Then Graeme McRae proved that Sum_{n >= 1} b(n)*x^n = Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 2} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = A000217(k) = k*(k+1)/2.

Since a(n) - b(n) = A000005(n) for n >= 1, to finish the proof, we only need to show that K(x) := 1 + Sum_{n >= 1} a(n)*x^n - Sum_{n >= 1} b(n)*x^n is the g.f. of A000005 (= number of divisors). But it is easy to show that K(x) = 1 + Sum_{k >= 1} x^k/(1 - x^k) = 1 + Sum_{n >= 1} A000005(n)*x^n (Lambert series for the number of divisors function). (End)

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

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

The Heinz numbers of these partitions are given by A325367.

The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.

In other words, a(n) is the number of oblong numbers (A002378) dividing n. - Bernard Schott, Jul 29 2022

Also the number of partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences. - Gus Wiseman, Mar 31 2020