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

%I A332626 #15 Feb 16 2025 08:33:59
%S A332626 1,2,4,20,40,152,664,1760,5680,24752,138064,356480,1568320,5886752,
%T A332626 32781664,266420000,726928960,3135277952,16299729664,81402739520,
%U A332626 640678081600,7084434124352,18897678264064,92846198695040,464088929482240,3347512310365952
%N A332626 Sum of least integers of prime signatures over all partitions of n into distinct parts.
%H A332626 Alois P. Heinz, <a href="/A332626/b332626.txt">Table of n, a(n) for n = 0..712</a>
%H A332626 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSignature.html">Prime Signature</a>
%H A332626 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A332626 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_signature">Prime signature</a>
%H A332626 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%F A332626 a(n) = Sum_{k=1..A000009(n)} A328524(n,k).
%e A332626 a(5) = 2^5 + 2^4*3^1 + 2^3*3^2 = 32 + 48 + 72 = 152.
%p A332626 b:= proc(n, i, j) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
%p A332626       1, b(n, i-1, j)+ithprime(j)^i*b(n-i, min(n-i, i-1), j+1)))
%p A332626     end:
%p A332626 a:= n-> b(n$2, 1):
%p A332626 seq(a(n), n=0..30);
%t A332626 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]]];
%t A332626 a[n_] := b[n, n, 1];
%t A332626 a /@ Range[0, 30] (* _Jean-François Alcover_, May 04 2020, after Maple *)
%Y A332626 Row sums of A328524.
%Y A332626 Cf. A000009, A025487, A074140.
%K A332626 nonn
%O A332626 0,2
%A A332626 _Alois P. Heinz_, Feb 17 2020