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A339382
Number of partitions of n into an even number of distinct primes (counting 1 as a prime).
4
1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 6, 5, 5, 5, 6, 6, 7, 7, 9, 8, 9, 8, 11, 10, 11, 12, 14, 12, 15, 14, 17, 16, 17, 17, 22, 20, 22, 21, 25, 24, 28, 27, 31, 30, 33, 31, 39, 36, 40, 40, 46, 42, 49, 47, 54, 53, 58, 55, 67, 63, 70, 68
OFFSET
0,9
FORMULA
G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) + (1 - x) * Product_{k>=1} (1 - x^prime(k))).
a(n) = (A036497(n) + A298602(n)) / 2.
EXAMPLE
a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].
MAPLE
s:= proc(n) option remember;
`if`(n<1, n+1, ithprime(n)+s(n-1))
end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
`if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..100); # Alois P. Heinz, Dec 02 2020
MATHEMATICA
nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] + (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 02 2020
STATUS
approved