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%I #4 Feb 06 2021 08:53:43
%S 1,1,2,3,5,6,10,13,19,24,34,42,58,71,94,116,151,182,234,282,354,424,
%T 528,627,773,914,1113,1311,1585,1854,2227,2599,3095,3597,4262,4931,
%U 5811,6704,7855,9035,10542,12080,14036,16047,18561,21161,24397,27736,31866
%N Number of partitions of 2*n into exactly n prime powers (including 1).
%F G.f.: Product_{p prime, k>=1} 1 / (1 - x^(p^k-1)).
%t nmax = 48; CoefficientList[Series[Product[1/(1 - Boole[PrimePowerQ[k + 1]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Boole[PrimePowerQ[d + 1]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]
%Y Cf. A000961, A023893, A181062, A280954, A341153.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Feb 06 2021