OFFSET
0,4
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+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 + 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(prod(k=1, N, 1+isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 10 2018
STATUS
approved