A184172 Number of partitions of n into an odd number of distinct primes.

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

Offset

0,19

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+1). - Alois P. Heinz, Nov 15 2012

a(n) + A184171(n) = A000586(n). - R. J. Mathar, Mar 31 2023

Example

a(33) = 4 because we have [23,7,3], [19,11,3], [17,13,3], and [17,11,5].

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], i=1..iquo(nops(l), 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]], {i, 1, Quotient[Length[l], 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)) - Vec(parts(n, isprime, -1)))/2} \\ Andrew Howroyd, Dec 28 2017

Crossrefs

Cf. A000586, A184171, A184199.

Sequence in context: A105068 A363274 A120676 * A125973 A227196 A189172

Adjacent sequences: A184169 A184170 A184171 * A184173 A184174 A184175

Keyword

nonn

Author

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

Status

approved