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

0,3

Links

Table of n, a(n) for n=0..48.

Formula

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

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}]

Crossrefs

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

Sequence in context: A087750 A341131 A288253 * A035959 A036801 A035966

Adjacent sequences: A341151 A341152 A341153 * A341155 A341156 A341157

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 06 2021

Status

approved