A184199 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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