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A341140 Number of partitions of n into 3 distinct prime powers (including 1).

%I A341140 #9 Jul 13 2021 09:19:42
%S A341140 1,1,2,2,3,3,5,5,7,6,8,7,8,8,10,10,12,11,12,12,13,12,16,15,15,16,18,
%T A341140 17,19,20,21,24,22,22,23,25,22,27,26,25,26,29,25,31,27,30,31,34,26,34,
%U A341140 31,35,32,38,29,40,32,36,34,41,29,44,35,41,36,47,34,51,38,45,41,54
%N A341140 Number of partitions of n into 3 distinct prime powers (including 1).
%p A341140 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341140 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341140       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341140       `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
%p A341140     end:
%p A341140 a:= n-> b(n$2, 3):
%p A341140 seq(a(n), n=6..77);  # _Alois P. Heinz_, Feb 05 2021
%t A341140 q[n_] := q[n] = PrimeNu[n] < 2;
%t A341140 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341140      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341140      If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
%t A341140 a[n_] := b[n, n, 3];
%t A341140 Table[a[n], {n, 6, 77}] (* _Jean-François Alcover_, Jul 13 2021, after _Alois P. Heinz_ *)
%Y A341140 Cf. A000961, A010055, A282064, A307727, A307825, A341112, A341132, A341141, A341142, A341143, A341144, A341145.
%K A341140 nonn
%O A341140 6,3
%A A341140 _Ilya Gutkovskiy_, Feb 05 2021