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A341123 Number of partitions of n into 5 prime powers (including 1).

%I A341123 #10 Feb 22 2022 03:36:52
%S A341123 1,1,2,3,5,6,9,11,15,17,23,25,32,34,42,45,55,56,68,71,83,84,100,100,
%T A341123 117,118,136,135,158,153,179,178,204,200,234,226,261,255,291,283,327,
%U A341123 310,357,344,390,371,430,405,466,444,505,476,550,511,589,557,634,589,684,629
%N A341123 Number of partitions of n into 5 prime powers (including 1).
%p A341123 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341123 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341123       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341123       `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
%p A341123     end:
%p A341123 a:= n-> b(n$2, 5):
%p A341123 seq(a(n), n=5..64);  # _Alois P. Heinz_, Feb 05 2021
%t A341123 q[n_] := q[n] = Length[FactorInteger[n]] < 2;
%t A341123 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341123      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341123      If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
%t A341123 a[n_] := b[n, n, 5];
%t A341123 Table[a[n], {n, 5, 64}] (* _Jean-François Alcover_, Feb 22 2022, after _Alois P. Heinz_ *)
%Y A341123 Cf. A000961, A010055, A071330, A341112, A341122, A341124, A341125, A341126, A341127.
%K A341123 nonn
%O A341123 5,3
%A A341123 _Ilya Gutkovskiy_, Feb 05 2021