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Expansion of exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^k ) * x^k/k).
1

%I #5 Sep 07 2018 04:47:19

%S 1,0,2,9,6,643,182,118953,6019,242630,2243190,25938251679,78106516,

%T 23349992199606,288964822371,46755212195033,226472341461312,

%U 48661337027901364945,18066374340919781,104224677113940850317679,440728415311733637734,208546898802899685866735,972477473959172989443327

%N Expansion of exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^k ) * x^k/k).

%F G.f.: Product_{k>=1} 1/(1 - prime(k)^prime(k)*x^prime(k))^(1/prime(k)).

%t nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

%t nmax = 22; CoefficientList[Series[Product[1/(1 - Prime[k]^Prime[k] x^Prime[k])^(1/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]

%Y Cf. A000607, A023881, A200768, A219224, A220427, A318913, A318968.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 06 2018