A280586 Expansion of Product_{p prime, k>=2} 1/(1 - x^(p^k)).

1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 4, 2, 1, 0, 4, 2, 1, 0, 6, 5, 2, 2, 6, 5, 2, 2, 10, 8, 5, 4, 12, 8, 5, 4, 16, 14, 8, 9, 18, 16, 8, 9, 24, 23, 15, 14, 30, 25, 18, 14, 36, 36, 26, 25, 42, 42, 29, 28, 52, 54, 42, 40, 65, 60, 50, 43, 78, 78, 65, 63, 93, 92, 73, 72, 110, 117, 96, 94, 135, 133, 114, 103, 158, 166, 145

Offset: 0

Author

Ilya Gutkovskiy, Jan 06 2017

Keywords

• nonn

Comments

Number of partitions of n into proper prime powers (A246547).

Examples

			a(16) = 4 because we have [16], [8, 8], [8, 4, 4] and [4, 4, 4, 4].
		

Links

• Eric Weisstein's World of Mathematics, Prime Power
• Index entries for related partition-counting sequences

Crossrefs

Cf. A023893, A023894, A246547.

Programs

Mathematica

nmax = 90; CoefficientList[Series[Product[1/(1 - Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

PARI

x='x+O('x^68); Vec(prod(k=2, 67, 1/(1 - sign(bigomega(k) - 1) * (1\omega(k)) * x^k))) \\ Indranil Ghosh, Apr 03 2017

Formula

G.f.: Product_{p prime, k>=2} 1/(1 - x^(p^k)).