OFFSET
1,2
COMMENTS
LINKS
Eric Weisstein's World of Mathematics, Prime Partition
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]]
Table[Total@Flatten[IntegerPartitions[n, All, Prime@Range@PrimePi@n]], {n, 52}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 10 2017
STATUS
approved