A184199 Number of partitions of n into an odd number of primes.

0, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 4, 3, 5, 4, 7, 6, 10, 8, 13, 11, 17, 15, 23, 20, 29, 26, 38, 34, 49, 43, 62, 55, 78, 69, 97, 88, 122, 109, 150, 135, 186, 167, 227, 205, 277, 251, 337, 306, 407, 371, 492, 448, 591, 539, 707, 647, 845, 773, 1005, 922, 1193, 1096, 1412, 1298, 1667, 1535, 1963, 1809, 2305, 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)=8 are odd, namely  11+5+2, 13+3+2, 5+5+3+3+2, 7+3+3+3+2, 7+5+2+2+2, 3+3+3+3+2+2+2, 5+3+2+2+2+2+2, 2+2+2+2+2+2+2+2+2.
The remaining A184198(18)=11 are even.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..200 from Jean-François Alcover)

Crossrefs

Cf. A000607, A048165, A184198, A184172.

Programs

Mathematica

a[n_] := a[n] = Count[IntegerPartitions[n, All, Prime[Range[PrimePi[n]]]], p_ /; OddQ[Length[p]]];
Reap[Do[Print[n, " ", a[n]]; Sow[a[n]], {n, 0, 200}]][[2, 1]] (* Jean-François Alcover, Feb 13 2020 *)

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)) - Vec(parts(n, isprime, -1)))/2} \\ Andrew Howroyd, Dec 28 2017

Formula

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

Extensions

a(31)-a(70) corrected by Andrew Howroyd, Dec 28 2017