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

0,17

Links

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

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

Example

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].

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 *)

Prog

(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

Crossrefs

Cf. A000586, A184172, A184198.

Sequence in context: A050252 A025877 A219182 * A362716 A133989 A029398

Adjacent sequences: A184168 A184169 A184170 * A184172 A184173 A184174

Keyword

nonn

Author

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

Status

approved