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%I #11 Dec 06 2021 16:17:19
%S 1,-1,1,3,1,5,1,3,10,9,1,9,1,13,16,3,1,14,1,13,22,21,1,9,26,25,37,17,
%T 1,30,1,3,34,33,36,18,1,37,40,13,1,40,1,25,70,45,1,9,50,34,52,29,1,41,
%U 56,17,58,57,1,34,1,61,94,3,66,60,1,37,70,58,1,18,1,73,116,41,78,70,1,13,118,81,1,44,86
%N L.g.f.: log(Product_{k>=1} (1 + x^k)/(1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.
%H Michael De Vlieger, <a href="/A300893/b300893.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: Sum_{k>=1} A018252(k)*x^A018252(k)/(1 + x^A018252(k)).
%F a(n) = 1 if n is an odd prime or 1 (A006005).
%e L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 + 5*x^6/6 + x^7/7 + 3*x^8/8 + 10*x^9/9 + 9*x^10/10 + ...
%e exp(L(x)) = 1 + x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 3*x^10 + ... + A096258(n)*x^n + ...
%t nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)/(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t nmax = 85; Rest[CoefficientList[Series[Sum[Boole[!PrimeQ[k]] k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t Table[DivisorSum[n, (-1)^(n/# + 1) # &, !PrimeQ[#] &], {n, 85}]
%Y Cf. A006005, A018252, A023890, A096258, A300852.
%K sign
%O 1,4
%A _Ilya Gutkovskiy_, Mar 14 2018
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