A341154 Number of partitions of 2*n into exactly n prime powers (including 1).

1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 71, 94, 116, 151, 182, 234, 282, 354, 424, 528, 627, 773, 914, 1113, 1311, 1585, 1854, 2227, 2599, 3095, 3597, 4262, 4931, 5811, 6704, 7855, 9035, 10542, 12080, 14036, 16047, 18561, 21161, 24397, 27736, 31866

Offset: 0

Author

Ilya Gutkovskiy, Feb 06 2021

Keywords

• nonn

Crossrefs

Cf. A000961, A023893, A181062, A280954, A341153.

Programs

Mathematica

nmax = 48; CoefficientList[Series[Product[1/(1 - Boole[PrimePowerQ[k + 1]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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}]

Formula

G.f.: Product_{p prime, k>=1} 1 / (1 - x^(p^k-1)).