A332626 Sum of least integers of prime signatures over all partitions of n into distinct parts.

1, 2, 4, 20, 40, 152, 664, 1760, 5680, 24752, 138064, 356480, 1568320, 5886752, 32781664, 266420000, 726928960, 3135277952, 16299729664, 81402739520, 640678081600, 7084434124352, 18897678264064, 92846198695040, 464088929482240, 3347512310365952

Offset: 0

Author

Alois P. Heinz, Feb 17 2020

Keywords

• nonn

Examples

			a(5) = 2^5 + 2^4*3^1 + 2^3*3^2 = 32 + 48 + 72 = 152.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..712
• Eric Weisstein's World of Mathematics, Prime Signature
• Wikipedia, Partition (number theory)
• Wikipedia, Prime signature
• Index entries for sequences related to prime signature

Crossrefs

Row sums of A328524.

Cf. A000009, A025487, A074140.

Programs

Maple

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..30);

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, 0, If[n == 0, 1, b[n, i - 1, j] + Prime[j]^i b[n - i, Min[n - i, i - 1], j + 1]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 30] (* Jean-François Alcover, May 04 2020, after Maple *)

Formula

a(n) = Sum_{k=1..A000009(n)} A328524(n,k).