cp's OEIS Frontend

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

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 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

(End)

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

Writing a partition of n in the form sum(k>=1, c(k) * k) another (in general different) partition is obtained as sum(k>=1, k * c(k)). For example, the partition 6 = 4* 1 + 1* 2 = 1 + 1 + 1 + 1 + 2 is mapped to 1* 4 + 2 *1 = 2* 1 + 1* 4 = 2 + 2 + 4. This sequence counts the fixed points of this map.

The map is not surjective. For example, all partitions into distinct parts are mapped to n* 1.

The map is an involution for partitions where the multiplicities of all parts are distinct (Wilf partitions, see A098859). If in addition the set of parts the same as the set of multiplicities then the partition is a fixed point.

The second part of the preceding comment is incorrect. For example, the partition (3,3,2,1,1,1) maps to (3,2,2,2,1,1) so is not a fixed point, even though the set of parts is identical to the set of multiplicities. - Gus Wiseman, May 04 2019

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325325.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The version where zero is prepended to the prime indices is A287352.

One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.