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A341112 Number of partitions of n into 3 prime powers (including 1).

%I A341112 #22 Feb 22 2022 03:50:35
%S A341112 1,1,2,3,4,4,6,6,8,8,10,9,12,10,13,12,15,13,17,15,18,15,19,16,21,17,
%T A341112 23,18,24,19,27,23,30,24,32,25,32,26,34,26,36,26,36,28,38,28,40,30,42,
%U A341112 32,43,30,45,32,47,35,49,30,50,35,51,36,53,35,55,37,54,40,57,36,61,40,61
%N A341112 Number of partitions of n into 3 prime powers (including 1).
%p A341112 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341112 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341112       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341112       `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
%p A341112     end:
%p A341112 a:= n-> b(n$2, 3):
%p A341112 seq(a(n), n=3..75);  # _Alois P. Heinz_, Feb 05 2021
%t A341112 q[n_] := q[n] = Length[FactorInteger[n]] < 2;
%t A341112 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341112      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341112      If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
%t A341112 a[n_] := b[n, n, 3];
%t A341112 Table[a[n], {n, 3, 75}] (* _Jean-François Alcover_, Feb 22 2022, after _Alois P. Heinz_ *)
%Y A341112 Cf. A000961, A010055, A071330, A282064, A307727, A307825, A341122, A341123, A341124, A341125, A341126, A341127.
%K A341112 nonn
%O A341112 3,3
%A A341112 _Ilya Gutkovskiy_, Feb 05 2021