A184198 Number of partitions of n into an even number of primes.

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

Author

R. J. Mathar, Jan 10 2011

Keywords

• nonn
• easy

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000607, A048165, A184199, A184171.

Programs

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 *)

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

Formula

a(n) = (A000607(n)+A048165(n))/2.

Extensions

a(31)-a(69) corrected by Andrew Howroyd, Dec 28 2017