A276560 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).

0, 2, 3, 4, 10, 12, 21, 24, 36, 50, 66, 84, 117, 140, 180, 224, 289, 342, 437, 520, 630, 770, 920, 1104, 1300, 1560, 1809, 2156, 2523, 2940, 3441, 3968, 4620, 5338, 6125, 7092, 8103, 9272, 10608, 12080, 13776, 15624, 17759, 20064, 22680, 25622, 28858, 32496, 36456, 40950, 45849, 51324, 57399, 64044, 71390

Offset: 1

Author

Ilya Gutkovskiy, Apr 10 2017

Keywords

• nonn

Comments

Sum of all parts of all partitions of n into prime parts.

Convolution of the sequences A000607 and A008472.

Examples

			a(6) = 12 because we have [3, 3], [2, 2, 2] and 2*6 = 12.
		

Links

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

Crossrefs

Cf. A000607, A008472, A066186, A084993.

Programs

Mathematica

nmax = 55; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}] Product[1/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 55; Rest[CoefficientList[Series[x D[Product[1/(1 - x^Prime[k]), {k, 1, nmax}], x], {x, 0, nmax}], x]]
Table[Total@Flatten[IntegerPartitions[n,All,Prime@Range@PrimePi@n]],{n,52}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

Formula

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

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

a(n) = n*A000607(n).

a(n) ~ n*exp(2*Pi*sqrt(n/log(n))/sqrt(3)).