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A276560 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Convolution of the sequences A000607 and A008472.

LINKS

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Prime Partition

Index entries for related partition-counting sequences

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)).

EXAMPLE

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

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]]

CROSSREFS

Cf. A000607, A008472, A066186, A084993.

Sequence in context: A229593 A196007 A134170 * A049548 A005456 A100773

Adjacent sequences:  A276557 A276558 A276559 * A276561 A276562 A276563

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 10 2017

STATUS

approved

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Last modified March 25 22:28 EDT 2019. Contains 321477 sequences. (Running on oeis4.)