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A300852 L.g.f.: log(Product_{k>=1} (1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

%I A300852 #10 Feb 05 2024 10:49:16
%S A300852 0,2,3,-2,5,-1,7,-2,3,-3,11,-5,13,-5,8,-2,17,-1,19,-7,10,-9,23,-5,5,
%T A300852 -11,3,-9,29,-6,31,-2,14,-15,12,-5,37,-17,16,-7,41,-8,43,-13,8,-21,47,
%U A300852 -5,7,-3,20,-15,53,-1,16,-9,22,-27,59,-10,61,-29,10,-2,18,-12,67,-19,26,-10,71,-5,73,-35,8
%N A300852 L.g.f.: log(Product_{k>=1} (1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.
%F A300852 G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)).
%F A300852 a(n) = Sum_{p|n, p prime} p * (-1)^(n/p + 1). [See Mmca prog.]
%e A300852 L.g.f.: L(x) = 2*x^2/2 + 3*x^3/3 - 2*x^4/4 + 5*x^5/5 - x^6/6 + 7*x^7/7 - 2*x^8/8 + 3*x^9/9 - 3*x^10/10 + ...
%e A300852 exp(L(x)) = 1 + x^2 + x^3 + 2*x^5 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... + A000586(n)*x^n + ...
%t A300852 nmax = 75; Rest[CoefficientList[Series[Log[Product[(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}],x] Range[0, nmax]]
%t A300852 nmax = 75; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A300852 Table[DivisorSum[n, (-1)^(n/# + 1) # &, PrimeQ[#] &], {n, 75}]
%Y A300852 Cf. A000040 (fixed points), A000586, A008472, A300893, A300894.
%K A300852 sign,easy
%O A300852 1,2
%A A300852 _Ilya Gutkovskiy_, Mar 13 2018