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

0,9

Links

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

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

Cf. A000607, A048165, A184199, A184171.

Sequence in context: A165419 A117660 A358015 * A238998 A355359 A144369

Adjacent sequences: A184195 A184196 A184197 * A184199 A184200 A184201

Keyword

nonn,easy

Author

R. J. Mathar, Jan 10 2011

Extensions

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

Status

approved