A338117 Number of partitions of n into two parts (s,t) such that (t-s) | n, where s < t.

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 2, 1, 3, 3, 1, 1, 5, 2, 1, 3, 3, 1, 3, 1, 4, 3, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 3, 3, 1, 3, 3, 5, 3, 1, 1, 7, 1, 1, 5, 5, 3, 3, 1, 3, 3, 3, 1, 8, 1, 1, 5, 3, 3, 3, 1, 7, 4, 1, 1, 7, 3, 1, 3, 5, 1, 5, 3, 3, 3

Offset: 1

Author

Wesley Ivan Hurt, Oct 10 2020

Keywords

• nonn
• easy

Comments

Apparently a(n) = A320111(n) - 1. - Hugo Pfoertner, Oct 30 2020

The above observation is true, which can be seen from the formula A320111(2n) = A000005(n), A320111(2n+1) = A000005(2n+1). For odd numbers, the difference (t-s) may range over all the divisors of n except the n itself, and for even numbers the difference (t-s) [which is always even] may range only over the even divisors of n, except the n itself. Note that A000005(2n) = A000005(n) + A001227(n). - Antti Karttunen, Dec 12 2021

Examples

			a(8) = 2; The partitions of 8 into two parts (s,t) such that s < t are (7,1), (6,2), (5,3) and (4,4). Only the partitions (6,2) and (5,3) have (6-2) | 8 and (5-3) | 8, so a(8) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20160
• Index entries for sequences related to partitions

Crossrefs

Cf. A000005, A001227, A320111.

Cf. also A337101, A338021.

Programs

Mathematica

Table[Sum[(1 - Ceiling[n/(n - 2 i)] + Floor[n/(n - 2 i)]), {i, Floor[(n - 1)/2]}], {n, 100}]

PARI

for(n=1,85,my(j=0);forpart(x=n,if(#x==2,if(x[2]!=x[1]&&!(n%(x[2]-x[1])),j++)));print1(j,", ")) \\ Hugo Pfoertner, Oct 30 2020

PARI

A338117(n) = sum(s=1,(n-1)\2,!(n%(n-(2*s)))); \\ Antti Karttunen, Dec 12 2021

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} (1 - ceiling(n/(n-2*i)) + floor(n/(n-2*i))).