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

0,8

Links

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

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)=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.

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 *)

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

Crossrefs

Cf. A000607, A048165, A184198, A184172.

Sequence in context: A341461 A337485 A328678 * A262346 A242560 A204596

Adjacent sequences: A184196 A184197 A184198 * A184200 A184201 A184202

Keyword

nonn,easy

Author

R. J. Mathar, Jan 10 2011

Extensions

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

Status

approved