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A219199 Number of partitions of n into 5 distinct primes.

%I A219199 #20 Apr 26 2025 08:28:09
%S A219199 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 A219199 6,14,10,16,9,18,11,19,15,21,14,22,16,25,22,26,20,29,25,31,29,32,29,
%U A219199 35,34,39,39,38,39,43,45,48,50,46,53,53,57,57,66,55
%N A219199 Number of partitions of n into 5 distinct primes.
%H A219199 Alois P. Heinz, <a href="/A219199/b219199.txt">Table of n, a(n) for n = 28..10000</a>
%F A219199 G.f.: Sum_{0<i_1<i_2<...<i_5} x^(Sum_{j=1..5} prime(i_j)).
%F A219199 a(n) = [x^n*y^5] Product_{i>=1} (1+x^prime(i)*y).
%p A219199 b:= proc(n, i) option remember; `if`(n=0, [1,0$5], `if`(i<1, [0$6],
%p A219199        zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$5],
%p A219199        b(n-ithprime(i), i-1)[1..5])[]], 0)))
%p A219199     end:
%p A219199 a:= n-> b(n, numtheory[pi](n))[6]:
%p A219199 seq(a(n), n=28..100);
%t A219199 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_ *)
%t A219199 Table[Length@Select[IntegerPartitions[k,{5}, Prime@Range@100], #[[1]] > #[[2]] > #[[3]] > #[[4]] > #[[5]] &], {k, 28, 100}] (* _Robert Price_, Apr 25 2025 *)
%Y A219199 Column k=5 of A219180.
%K A219199 nonn
%O A219199 28,7
%A A219199 _Alois P. Heinz_, Nov 14 2012