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A339396
Number of partitions of n into an odd number of nonprime parts.
1
0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 6, 6, 9, 9, 13, 14, 18, 20, 26, 29, 37, 39, 51, 57, 69, 78, 95, 105, 129, 141, 173, 192, 231, 255, 306, 340, 403, 446, 531, 585, 691, 764, 896, 995, 1160, 1279, 1493, 1652, 1911, 2117, 2443, 2700, 3109, 3434, 3941, 4357, 4983, 5496, 6277
OFFSET
0,7
FORMULA
G.f.: (1/2) * (Product_{k>=1} (1 - x^prime(k)) / (1 - x^k) - Product_{k>=1} (1 + x^prime(k)) / (1 + x^k)).
a(n) = (A002095(n) - A302236(n)) / 2.
EXAMPLE
a(9) = 3 because we have [9], [4, 4, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].
MATHEMATICA
nmax = 60; CoefficientList[Series[(1/2) (Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 02 2020
STATUS
approved