%I
%S 2,3,9,11,12,14,17,18,19,20,28,44,47,51,52,55,56,58,59,62,64,65,69,80,
%T 81,82,83,87,91,92,94,96,99,105,106,107,113,118,119,126,127,131,147,
%U 155,157,160,161,162,164,178,179,180,215,218,224,227,257,259,269,295
%N Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1  x^j) / (1  x^prime(j)) is zero.
%C Numbers k such that number of partitions of k into an even number of distinct nonprime parts equals number of partitions of k into an odd number of distinct nonprime parts.
%C Positions of 0's in A302234.
%t a[j_] := a[j] = If[j == 0, 1, Sum[Sum[Boole[!PrimeQ[d]] d, {d, Divisors[k]}] a[j  k], {k, 1, j}]/j]; Select[Range[300], a[#] == 0 &]
%t Flatten[Position[nmax = 300; Rest[CoefficientList[Series[Product[(1  x^j)/(1  x^Prime[j]), {j, 1, nmax}], {x, 0, nmax}], x]], 0]]
%Y Cf. A002095, A018252, A090864, A096258, A302234.
%K nonn
%O 1,1
%A _Ilya Gutkovskiy_, Nov 01 2019
