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A328970 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. 0

%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

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Last modified August 8 14:04 EDT 2020. Contains 336298 sequences. (Running on oeis4.)