A219199 - OEIS

%I #14 Sep 21 2015 19:36:42

%S 1,0,1,0,0,0,2,0,2,0,2,1,4,0,4,1,4,2,6,1,6,2,6,4,8,2,10,5,9,6,11,5,13,

%T 6,14,10,16,9,18,11,19,15,21,14,22,16,25,22,26,20,29,25,31,29,32,29,

%U 35,34,39,39,38,39,43,45,48,50,46,53,53,57,57,66,55

%N Number of partitions of n into 5 distinct primes.

%H Alois P. Heinz, <a href="/A219199/b219199.txt">Table of n, a(n) for n = 28..10000</a>

%F G.f.: Sum_{0<i_1<i_2<...<i_5} x^(Sum_{j=1..5} prime(i_j)).

%F a(n) = [x^n*y^5] Product_{i>=1} (1+x^prime(i)*y).

%p b:= proc(n, i) option remember; `if`(n=0, [1,0$5], `if`(i<1, [0$6],

%p        zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$5],

%p        b(n-ithprime(i), i-1)[1..5])[]], 0)))

%p     end:

%p a:= n-> b(n, numtheory[pi](n))[6]:

%p seq(a(n), n=28..100);

%t k = 5; b[n_, i_] := b[n, i] = If[n == 0, Join[{1}, Array[0&, k]], If[i<1, Array[0&, k+1], Plus @@ PadRight[{b[n, i-1], Join[{0}, If[Prime[i]>n, Array[0&, k], Take[b[n-Prime[i], i-1], k]]]}]]]; a[n_] := b[n, PrimePi[n]][[k+1]]; Table[a[n], {n, 28, 100}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)

%Y Column k=5 of A219180.

%K nonn

%O 28,7

%A _Alois P. Heinz_, Nov 14 2012