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%I #15 Sep 17 2023 17:56:12
%S 1,0,0,0,1,0,2,0,1,0,2,0,4,0,2,1,5,1,6,0,5,2,6,1,10,1,9,4,11,3,15,3,
%T 14,6,16,6,22,5,20,10,25,11,29,9,29,16,34,17,39,15,39,25,45,24,50,25,
%U 53,35,57,34,66,36,68,48,75,50,83,52,88,65,92,69,104
%N Number of partitions of n into 6 distinct primes.
%H Alois P. Heinz, <a href="/A219200/b219200.txt">Table of n, a(n) for n = 41..10000</a>
%F G.f.: Sum_{0<i_1<i_2<...<i_6} x^(Sum_{j=1..6} prime(i_j)).
%F a(n) = [x^n*y^6] Product_{i>=1} (1+x^prime(i)*y).
%p b:= proc(n, i) option remember; `if`(n=0, [1,0$6], `if`(i<1, [0$7],
%p zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$6],
%p b(n-ithprime(i), i-1)[1..6])[]], 0)))
%p end:
%p a:= n-> b(n, numtheory[pi](n))[7]:
%p seq(a(n), n=41..120);
%t k = 6; 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, 41, 120}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)
%t Table[Count[IntegerPartitions[n,{6}],_?(AllTrue[#,PrimeQ]&&Length[Union[#]]==6&)],{n,41,120}] (* _Harvey P. Dale_, Sep 17 2023 *)
%Y Column k=6 of A219180.
%K nonn
%O 41,7
%A _Alois P. Heinz_, Nov 14 2012
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