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A341127 Number of partitions of n into 9 prime powers (including 1).

%I A341127 #9 Feb 22 2022 03:35:58
%S A341127 1,1,2,3,5,6,10,13,19,24,33,40,54,64,83,99,125,144,180,206,250,284,
%T A341127 341,383,455,506,593,656,762,835,965,1054,1206,1309,1491,1610,1825,
%U A341127 1964,2213,2374,2664,2843,3179,3387,3769,3998,4440,4695,5194,5480,6043,6357
%N A341127 Number of partitions of n into 9 prime powers (including 1).
%p A341127 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341127 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341127       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341127       `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
%p A341127     end:
%p A341127 a:= n-> b(n$2, 9):
%p A341127 seq(a(n), n=9..60);  # _Alois P. Heinz_, Feb 05 2021
%t A341127 q[n_] := q[n] = Length[FactorInteger[n]] < 2;
%t A341127 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341127      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341127      If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
%t A341127 a[n_] := b[n, n, 9];
%t A341127 Table[a[n], {n, 9, 60}] (* _Jean-François Alcover_, Feb 22 2022, after _Alois P. Heinz_ *)
%Y A341127 Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341124, A341125, A341126.
%K A341127 nonn
%O A341127 9,3
%A A341127 _Ilya Gutkovskiy_, Feb 05 2021