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%I #7 Jan 09 2019 09:18:31
%S 1,0,1,2,15,44,475,2274,33313,227240,2920041,26754650,469513231,
%T 4613913732,85842524755,1174844041994,24672317426625,334246510927184,
%U 7985602649948113,127351500133158450,3282809137540001551,60776696924693716700,1556379682561575238731,32568139442090869594802
%N Expansion of e.g.f. Product_{p prime, k>=1} 1/(1 - x^(p^k))^(1/(p^k)).
%F E.g.f.: exp(Sum_{k>=1} bigomega(k)*x^k/k), where bigomega(k) = number of prime divisors of k counted with multiplicity (A001222).
%p seq(n!*coeff(series(exp(add(bigomega(k)*x^k/k,k=1..100)),x=0,24),x,n),n=0..23); # _Paolo P. Lava_, Jan 09 2019
%t nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(Boole[PrimePowerQ[k]]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 23; CoefficientList[Series[Exp[Sum[PrimeOmega[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = (n - 1)! Sum[PrimeOmega[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
%Y Cf. A001222, A023894, A028342, A206303, A318912, A318913.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Sep 05 2018
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