A356919 Number of partitions of n into 5 parts that divide n.

0, 0, 0, 0, 1, 1, 0, 2, 1, 2, 0, 8, 0, 1, 3, 3, 0, 6, 0, 6, 1, 0, 0, 20, 1, 0, 1, 2, 0, 14, 0, 3, 0, 0, 1, 20, 0, 0, 0, 11, 0, 8, 0, 0, 5, 0, 0, 26, 0, 2, 0, 0, 0, 7, 1, 4, 0, 0, 0, 41, 0, 0, 2, 3, 1, 2, 0, 0, 0, 5, 0, 35, 0, 0, 3, 0, 0, 2, 0, 12, 1, 0, 0, 25, 1, 0, 0, 2, 0, 23, 0, 0, 0, 0, 1, 27, 0, 1, 1, 7, 0, 1, 0, 2, 4

Offset: 1

Author

Wesley Ivan Hurt, Sep 04 2022

Keywords

• nonn

Examples

			a(12) = 8; there are 8 ways to write 12 as the sum of 5 divisors of 12: 6+3+1+1+1 = 6+2+2+1+1 = 4+4+2+1+1 = 4+3+3+1+1 = 4+3+2+2+1 = 4+2+2+2+2 = 3+3+3+2+1 = 3+3+2+2+2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to partitions

Crossrefs

Cf. A355641.

Programs

PARI

A356919(n, x=n, y=n, nparts=5) = if(0==nparts, (0==y), if(y<=0, 0, sumdiv(n, d, if((d<=x), A356919(n, d, y-d, nparts-1))))); \\ Antti Karttunen, Jan 13 2025

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(n/l) * c(n/k) * c(n/j) * c(n/i) * c(n/(n-i-j-k-l)), where c(n) = 1 - ceiling(n) + floor(n).

Extensions

More terms from Antti Karttunen, Jan 13 2025