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A341122 Number of partitions of n into 4 prime powers (including 1).

%I A341122 #12 Feb 22 2022 03:50:22
%S A341122 1,1,2,3,5,5,8,9,12,13,17,17,22,22,26,27,33,31,39,38,44,43,51,47,58,
%T A341122 54,63,60,71,64,79,74,88,82,99,88,108,97,116,105,126,110,134,119,141,
%U A341122 126,153,133,164,143,172,149,184,155,194,168,204,173,215,180,227,192,238
%N A341122 Number of partitions of n into 4 prime powers (including 1).
%p A341122 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341122 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341122       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341122       `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
%p A341122     end:
%p A341122 a:= n-> b(n$2, 4):
%p A341122 seq(a(n), n=4..66);  # _Alois P. Heinz_, Feb 05 2021
%t A341122 q[n_] := q[n] = Length[FactorInteger[n]] < 2;
%t A341122 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341122      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341122      If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
%t A341122 a[n_] := b[n, n, 4];
%t A341122 Table[a[n], {n, 4, 66}] (* _Jean-François Alcover_, Feb 22 2022, after _Alois P. Heinz_ *)
%Y A341122 Cf. A000961, A010055, A071330, A341112, A341123, A341124, A341125, A341126, A341127.
%K A341122 nonn
%O A341122 4,3
%A A341122 _Ilya Gutkovskiy_, Feb 05 2021