A038499 Number of partitions of n into a prime number of parts.

1, 0, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 39, 52, 65, 84, 104, 134, 165, 210, 258, 324, 397, 495, 603, 747, 908, 1115, 1351, 1652, 1993, 2425, 2918, 3531, 4237, 5106, 6105, 7330, 8741, 10449, 12425, 14804, 17549, 20839, 24637, 29155, 34377, 40559, 47688, 56100

Offset: 0

Author

Christian G. Bower, Feb 15 1999

Keywords

• nonn

Comments

Also, number of partitions of n whose largest part is a prime. E.g., for a(7) = 10 we have 6+1 = 5+2 = 4+3 = 5+1+1 = 4+2+1 = 3+3+1 = 3+2+2 = 3+1+1+1+1 = 2+2+1+1+1 = 1+1+1+1+1+1+1 and 7 = 5+2 = 5+1+1 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1. - Jon Perry Jul 06 2004

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A027187, A027193.

Programs

Maple

with(numtheory):
b:= proc(n, i) option remember; `if`(n<0, 0,
      `if`(n=0 or i=1, 1, `if`(i<1, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i)))))
    end:
a:= n-> `if`(n=0, 1, add((p-> b(n-p, p)
           )(ithprime(i)), i=1..pi(n))):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 24 2015

Mathematica

nn=50;Table[CoefficientList[Series[x^p Product[1/(1-x^i),{i,1,p}],{x,0,nn}],x],{p,Table[Prime[m],{m,1,PrimePi[nn]}]}]//Total  (* Geoffrey Critzer, Mar 10 2013 *)

Formula

G.f.: Sum_{n>=1}(x^prime(n)/Product_{i=1..prime(n)}(1-x^i)). - Vladeta Jovovic, Dec 25 2003