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A184198
Number of partitions of n into an even number of primes.
11
1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 6, 5, 8, 7, 11, 10, 15, 13, 20, 17, 26, 23, 34, 29, 43, 38, 55, 49, 69, 62, 88, 78, 109, 97, 135, 122, 167, 150, 205, 186, 251, 227, 306, 277, 371, 337, 448, 407, 539, 492, 647, 591, 773, 707, 922, 845, 1096, 1005, 1298, 1193, 1535, 1412, 1809, 1667, 2127
OFFSET
0,9
LINKS
FORMULA
a(n) = (A000607(n)+A048165(n))/2.
EXAMPLE
n=18 can be partitioned in A000607(18)=19 ways into primes, of which a(18)=11 are even, namely 11+7, 13+5, 5+5+5+3, 7+5+3+3, 3+3+3+3+3+3, 7+7+2+2, 11+3+2+2, 5+3+3+3+2+2, 5+5+2+2+2+2, 7+3+2+2+2+2, 3+3+2+2+2+2+2+2.
The remaining A184199(18)=8 are odd.
MATHEMATICA
Table[Count[IntegerPartitions[n], _?(AllTrue[#, PrimeQ]&&EvenQ[Length[ #]]&)], {n, 0, 70}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 16 2018 *)
PROG
(PARI)
parts(n, pred, y)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=80); Vec(parts(n, isprime, 1) + parts(n, isprime, -1))/2} \\ Andrew Howroyd, Dec 28 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jan 10 2011
EXTENSIONS
a(31)-a(69) corrected by Andrew Howroyd, Dec 28 2017
STATUS
approved