A280245 Expansion of Product_{k>=1} (1 + x^prime(k))^2.

1, 0, 2, 2, 1, 6, 1, 8, 6, 6, 14, 6, 18, 14, 18, 24, 23, 30, 35, 38, 46, 54, 55, 74, 72, 90, 100, 106, 128, 136, 152, 178, 185, 216, 238, 252, 302, 308, 359, 390, 420, 478, 512, 564, 628, 668, 745, 810, 871, 974, 1035, 1140, 1238, 1336, 1459, 1586, 1700, 1868, 1993, 2168, 2354, 2512, 2751, 2930, 3177, 3418, 3677, 3960

Offset: 0

Author

Ilya Gutkovskiy, Dec 29 2016

Keywords

• nonn

Comments

Number of partitions of n into distinct prime parts, with 2 types of each part.

Self-convolution of A000586. - Ilya Gutkovskiy, Jan 19 2018

Examples

			a(5) = 6 because we have [5], [5'], [3, 2], [3', 2], [3, 2'], [3', 2'].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
• Eric Weisstein's World of Mathematics, Prime Partition
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000009, A000586, A000607, A022567, A070215.

Programs

Mathematica

nmax = 67; CoefficientList[Series[Product[(1 + x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^prime(k))^2.

log(a(n)) ~ 2*Pi*sqrt(n/(3*log(n/2))). - Vaclav Kotesovec, Jan 12 2021