A038499 - OEIS

%I #36 Jan 02 2017 02:36:46

%S 1,0,1,2,3,5,7,10,13,18,23,31,39,52,65,84,104,134,165,210,258,324,397,

%T 495,603,747,908,1115,1351,1652,1993,2425,2918,3531,4237,5106,6105,

%U 7330,8741,10449,12425,14804,17549,20839,24637,29155,34377,40559,47688,56100

%N Number of partitions of n into a prime number of parts.

%C 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

%H Alois P. Heinz, <a href="/A038499/b038499.txt">Table of n, a(n) for n = 0..10000</a>

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

%p with(numtheory):

%p b:= proc(n, i) option remember; `if`(n<0, 0,

%p       `if`(n=0 or i=1, 1, `if`(i<1, 0, b(n, i-1)+

%p       `if`(i>n, 0, b(n-i, i)))))

%p     end:

%p a:= n-> `if`(n=0, 1, add((p-> b(n-p, p)

%p            )(ithprime(i)), i=1..pi(n))):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 24 2015

%t 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 *)

%Y Cf. A027187, A027193.

%K nonn

%O 0,4

%A _Christian G. Bower_, Feb 15 1999