A055923 Number of partitions of n in which each part occurs a prime number (or 0) times.

1, 0, 1, 1, 1, 1, 3, 2, 3, 4, 4, 6, 8, 8, 10, 13, 13, 20, 20, 24, 26, 38, 35, 51, 51, 65, 67, 92, 86, 121, 117, 153, 155, 209, 197, 270, 262, 339, 341, 444, 425, 565, 555, 703, 711, 903, 884, 1135, 1128, 1397, 1430, 1766, 1757, 2193, 2214, 2691, 2762, 3344

Offset

0,7

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Formula

EULER transform of b where b has g.f. Sum {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of prime numbers.

G.f.: Product(1+Sum(x^(i*prime(k)), k=1..infinity), i=1..infinity). - Vladeta Jovovic, Jan 08 2005

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(isprime(j), b(n-i*j, i-1), 0), j=1..n/i) +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[PrimeQ[j], b[n-i*j, i-1], 0], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A007690, A055922.

Sequence in context: A075988 A029150 A255246 * A035634 A091563 A161985

Adjacent sequences: A055920 A055921 A055922 * A055924 A055925 A055926

Keyword

nonn

Author

Christian G. Bower, Jun 23 2000

Status

approved