A302234 -id:A302234 - cp's OEIS frontend

A302236 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 + x^k).

1, -1, 1, -1, 0, 0, -1, 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 1, 0, 0, 0, -1, 1, 0, -1, 1, -2, 1, 0, 0, 2, -1, 0, 0, -1, 2, -1, -1, 1, -2, 1, 0, 0, 0, -2, -1, 2, 0, 0, 1, -3, 2, -1, 1, 2, -2, -1, -1, 1, 3, 0, -2, 1, -2, 0, 3, 0, 0, -2, -2, 5, 1, 1, -1, -4, 1, -1, 2, 4, -2

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

sign

The difference between the number of partitions of n into an even number of nonprime parts and the number of partitions of n into an odd number of nonprime parts.

Convolution of the sequences A000586 and A081362.

Index entries for sequences related to partitions

Cf. A000586, A002095, A018252, A048165, A081362, A096258, A302234.

nmax = 80; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[1/(1 + Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 + x^A018252(k)).

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.

Original entry on oeis.org

2, 3, 9, 11, 12, 14, 17, 18, 19, 20, 28, 44, 47, 51, 52, 55, 56, 58, 59, 62, 64, 65, 69, 80, 81, 82, 83, 87, 91, 92, 94, 96, 99, 105, 106, 107, 113, 118, 119, 126, 127, 131, 147, 155, 157, 160, 161, 162, 164, 178, 179, 180, 215, 218, 224, 227, 257, 259, 269, 295

Offset: 1

Ilya Gutkovskiy, Nov 01 2019

nonn

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.

Positions of 0's in A302234.

Cf. A002095, A018252, A090864, A096258, A302234.

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 &]
Flatten[Position[nmax = 300; Rest[CoefficientList[Series[Product[(1 - x^j)/(1 - x^Prime[j]), {j, 1, nmax}], {x, 0, nmax}], x]], 0]]

cp's OEIS Frontend

A302236 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 + x^k).

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