A184171 Number of partitions of n into an even number of distinct primes.

1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 5, 5, 4, 6, 5, 5, 6, 7, 7, 8, 7, 9, 8, 9, 8, 11, 11, 12, 10, 13, 12, 14, 14, 15, 16, 17, 16, 20, 19, 20, 20, 24, 22, 26, 23, 27, 27, 30, 28, 34, 33, 36, 34, 40, 37, 43, 41, 46, 46, 50, 47, 56, 55

Offset: 0

Author

Emeric Deutsch, (suggested by R. J. Mathar), Jan 09 2011

Keywords

• nonn

Examples

			a(33) = 5 because we have [31,2], [23,5,3,2], [19,7,5,2], [17,11,3,2], and [13,11,7,2].
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Alois P. Heinz)

Crossrefs

Cf. A000586, A184172, A184198.

Programs

Maple

g := 1/2*(Product(1+z^ithprime(k), k = 1 .. 120)+Product(1-z^ithprime(k), k = 1 .. 120)): gser := series(g, z = 0, 110): seq(coeff(gser, z, n), n = 0 .. 85);
# second Maple program
with(numtheory):
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, pi(n));
      add(l[2*i-1], i=1..iquo(nops(l)+1,2))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]}]]]; a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[2*i-1]], {i, 1, Quotient[Length[l]+1, 2]}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

PARI

parts(n, pred, y)={prod(k=1, n, 1 + if(pred(k), y*x^k + O(x*x^n), 0))}
{my(n=80); Vec(parts(n, isprime, 1) + parts(n, isprime, -1))/2} \\ Andrew Howroyd, Dec 28 2017

Formula

G.f.: (1/2)*[Product_{k>=1} (1+z^prime(k)) + Product_{k>=1} (1-z^prime(k))].

a(n) = Sum_{k>=0} A219180(n,2*k). - Alois P. Heinz, Nov 15 2012

a(n) + A184172(n) = A000586(n). - R. J. Mathar, Mar 31 2023