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A339380
Number of partitions of n into an even number of primes (counting 1 as a prime).
5
1, 0, 1, 1, 3, 2, 5, 4, 9, 7, 14, 11, 22, 18, 33, 27, 48, 40, 69, 58, 97, 82, 134, 114, 183, 157, 246, 212, 327, 284, 431, 376, 562, 493, 728, 640, 934, 825, 1191, 1056, 1508, 1341, 1899, 1694, 2377, 2126, 2960, 2654, 3668, 3297, 4523, 4075, 5554, 5015, 6792, 6145
OFFSET
0,5
FORMULA
G.f.: (1/2) * ((1/(1 - x)) * Product_{k>=1} 1 / (1 - x^prime(k)) + (1/(1 + x)) * Product_{k>=1} 1 / (1 + x^prime(k))).
a(n) = (A034891(n) + A338826(n)) / 2.
EXAMPLE
a(6) = 5 because we have [5, 1], [3, 3], [3, 1, 1, 1], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].
MAPLE
b:= proc(n, i, t) option remember; (p->
`if`(n=0, t, `if`(i<0, 0, b(n, i-1, t)+
`if`(p>n, 0, b(n-p, i, 1-t)))))(`if`(i<1, 1, ithprime(i)))
end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..60); # Alois P. Heinz, Dec 02 2020
MATHEMATICA
nmax = 55; CoefficientList[Series[(1/2) ((1/(1 - x)) Product[1/(1 - x^Prime[k]), {k, 1, nmax}] + (1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
Table[Count[(Boole[PrimeQ/@(IntegerPartitions[n]/.(1->2))]), _?(EvenQ[Length[#]] && FreeQ[ #, 0]&)], {n, 0, 60}] (* Harvey P. Dale, Aug 20 2024 *)
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 02 2020
STATUS
approved