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

0,4

Links

Table of n, a(n) for n=0..24.

Formula

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

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}]

Prog

(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

Crossrefs

Cf. A002098, A007841, A319113.

Sequence in context: A135815 A055564 A077259 * A371493 A342986 A136589

Adjacent sequences: A319109 A319110 A319111 * A319113 A319114 A319115

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 10 2018

Status

approved