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%I #17 Dec 02 2020 09:00:16
%S 1,-1,0,-1,2,-2,2,-3,4,-4,5,-7,8,-9,11,-13,15,-18,21,-24,28,-32,37,
%T -43,49,-55,63,-72,81,-92,104,-117,131,-147,166,-185,206,-231,257,
%U -285,317,-353,391,-432,478,-528,583,-643,708,-778,855,-940,1031,-1130,1238,-1354
%N G.f.: (1/(1 + x)) * Product_{k>=1} 1/(1 + x^prime(k)).
%C The difference between the number of partitions of n into an even number of prime parts (including 1) and the number of partitions of n into an odd number of prime parts (including 1).
%C Convolution inverse of A036497.
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * A048165(k).
%t nmax = 55; CoefficientList[Series[(1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[k, (-1)^(k/#) # &, PrimeQ[#] || # == 1 &] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 55}]
%Y Cf. A000586, A000607, A008578, A036497, A046675, A048165, A298602.
%K sign
%O 0,5
%A _Ilya Gutkovskiy_, Dec 02 2020
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