A319112 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k)/prime(k)).

1, 0, 1, 2, 6, 44, 170, 1644, 7448, 72624, 653112, 8510160, 62704752, 1324662624, 10772812752, 167386388064, 2413326453120, 52610523489024, 597065112874368, 18066985168806144, 212119023906342144, 4734822914239173120, 100734270778298352384, 2818116390408742291968, 48201015565806837709824

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• nonn

Crossrefs

Cf. A002098, A007841, A319113.

Programs

Maple

seq(n!*coeff(series(mul(1/(1-x^ithprime(k)/ithprime(k)),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 24; CoefficientList[Series[Product[1/(1 - x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[PrimeQ[d]] d^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^(1-k/p) ) * x^k/k).