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A339395
Number of partitions of n into an even number of nonprime parts.
1
1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 8, 10, 13, 14, 19, 20, 26, 29, 36, 40, 51, 56, 70, 76, 96, 105, 129, 143, 172, 192, 231, 254, 308, 339, 402, 447, 529, 586, 691, 764, 896, 993, 1159, 1281, 1493, 1652, 1912, 2114, 2445, 2699, 3110, 3436, 3939, 4356, 4982, 5497, 6280
OFFSET
0,8
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 [8, 1], [6, 1, 1, 1] and [4, 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