A219198 - OEIS

%I #14 Sep 21 2015 19:36:17

%S 1,0,0,0,1,0,2,0,2,1,2,1,3,0,3,2,4,2,4,2,5,4,5,4,6,4,6,6,6,6,6,6,9,8,

%T 8,10,8,9,11,11,11,13,10,14,13,16,13,18,12,19,14,21,15,22,13,25,18,26,

%U 17,29,14,31,21,32,19,35,17,39,25,38,20,43,21,48,26

%N Number of partitions of n into 4 distinct primes.

%H Alois P. Heinz, <a href="/A219198/b219198.txt">Table of n, a(n) for n = 17..10000</a>

%F G.f.: Sum_{0<i_1<i_2<i_3<i_4} x^(Sum_{j=1..4} prime(i_j)).

%F a(n) = [x^n*y^4] Product_{i>=1} (1+x^prime(i)*y).

%p b:= proc(n, i) option remember; `if`(n=0, [1,0$4], `if`(i<1, [0$5],

%p        zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$4],

%p        b(n-ithprime(i), i-1)[1..4])[]], 0)))

%p     end:

%p a:= n-> b(n, numtheory[pi](n))[5]:

%p seq(a(n), n=17..100);

%t k = 4; 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, 17, 100}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)

%Y Column k=4 of A219180.

%K nonn

%O 17,7

%A _Alois P. Heinz_, Nov 14 2012