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%I #12 Feb 27 2022 12:02:19
%S 1,0,1,2,6,44,170,1644,7448,72624,653112,8510160,62704752,1324662624,
%T 10772812752,167386388064,2413326453120,52610523489024,
%U 597065112874368,18066985168806144,212119023906342144,4734822914239173120,100734270778298352384,2818116390408742291968,48201015565806837709824
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k)/prime(k)).
%F E.g.f.: exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^(1-k/p) ) * x^k/k).
%p 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
%t nmax = 24; CoefficientList[Series[Product[1/(1 - x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t 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]!
%t 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}]
%o (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
%Y Cf. A002098, A007841, A319113.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Sep 10 2018
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