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sumparts(n, pred)={sum(k=1, n, 1 - 1/(1+pred(k)*x^k) + O(x*x^n))*prod(k=1, n, 1+pred(k)*x^k + O(x*x^n))}
{my(n=60); Vec(sumparts(n, isprime), -n)} \\ Andrew Howroyd, Dec 28 2017
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G.f.: sum(x^p(j)/(1+x^p(j)),j>=1..infinity)*product(1+x^p(j), j>=1..infinity), where p(j) is the j-th prime. - Vladeta Jovovic, Jul 17 2003
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b[n_, i_] := b[n, i] = Module[{g}, If[n==0, {1, 0}, If[i < 1, {0, 0}, g = If[ Prime[i] > n, {0, 0}, b[n - Prime[i], i-1]]; b[n, i-1] + g + {0, g[[1]]}]]]; a[n_] := b[n, PrimePi[n]][[2]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 27 2015, after Alois P. Heinz *)
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G.f.: sum(x^p(j)/(1+x^p(j)),j=1..infinity)*product(1+x^p(j), j=1..infinity), where p(j) is the j-th prime. - _Vladeta Jovovic_
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