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A341132 Number of partitions of n into 2 distinct prime powers (including 1).

%I A341132 #21 Jul 13 2021 09:19:36
%S A341132 1,1,2,2,2,2,3,3,3,4,3,3,3,3,3,4,3,5,4,3,2,5,3,4,4,5,3,6,3,6,5,6,4,7,
%T A341132 2,5,4,6,3,6,3,6,5,5,2,8,3,7,4,6,2,8,3,7,4,5,2,8,3,6,4,6,3,9,2,8,5,7,
%U A341132 2,10,3,7,6,7,3,9,2,9,4,6,4,11,3,8,4,7,3,12
%N A341132 Number of partitions of n into 2 distinct prime powers (including 1).
%p A341132 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341132 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341132       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341132       `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
%p A341132     end:
%p A341132 a:= n-> b(n$2, 2):
%p A341132 seq(a(n), n=3..90);  # _Alois P. Heinz_, Feb 05 2021
%t A341132 q[n_] := q[n] = PrimeNu[n] < 2;
%t A341132 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341132      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341132      If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
%t A341132 a[n_] := b[n, n, 2];
%t A341132 Table[a[n], {n, 3, 90}] (* _Jean-François Alcover_, Jul 13 2021, after _Alois P. Heinz_ *)
%Y A341132 Cf. A000961, A010055, A071330, A282062, A306433, A307726, A341140, A341141, A341142, A341143, A341144, A341145.
%K A341132 nonn
%O A341132 3,3
%A A341132 _Ilya Gutkovskiy_, Feb 05 2021