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