%I #13 Sep 21 2015 19:37:56

%S 1,0,1,0,0,0,2,0,1,0,1,0,4,0,3,0,3,1,6,0,6,1,5,1,10,0,11,2,9,3,16,1,

%T 17,3,15,5,25,4,24,5,25,10,35,6,34,10,36,15,48,10,50,17,52,23,65,17,

%U 69,27,70,32,89,30,93,38,93,48,116,43,121,56,125,70,148

%N Number of partitions of n into 7 distinct primes.

%H Alois P. Heinz, <a href="/A219201/b219201.txt">Table of n, a(n) for n = 58..10000</a>

%F G.f.: Sum_{0<i_1<i_2<...<i_7} x^(Sum_{j=1..7} prime(i_j)).

%F a(n) = [x^n*y^7] Product_{i>=1} (1+x^prime(i)*y).

%p b:= proc(n, i) option remember; `if`(n=0, [1,0$7], `if`(i<1, [0$8],

%p zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$7],

%p b(n-ithprime(i), i-1)[1..7])[]], 0)))

%p end:

%p a:= n-> b(n, numtheory[pi](n))[8]:

%p seq(a(n), n=58..140);

%t k = 7; 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, 58, 140}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)

%Y Column k=7 of A219180.

%K nonn

%O 58,7

%A _Alois P. Heinz_, Nov 14 2012