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A219204 Number of partitions of n into 10 distinct primes.

%I A219204 #17 Apr 26 2025 08:28:24
%S A219204 1,0,1,0,0,0,0,0,2,0,0,0,2,0,3,0,1,0,4,0,5,0,3,0,7,0,9,0,7,1,10,0,16,
%T A219204 0,9,1,18,1,25,1,16,2,30,2,35,1,25,4,45,3,53,2,45,8,62,4,79,6,67,14,
%U A219204 90,8,112,10,96,19,126,16,158,17,135,29,182,26,210
%N A219204 Number of partitions of n into 10 distinct primes.
%H A219204 Alois P. Heinz, <a href="/A219204/b219204.txt">Table of n, a(n) for n = 129..10000</a>
%F A219204 G.f.: Sum_{0<i_1<i_2<...<i_10} x^(Sum_{j=1..10} prime(i_j)).
%F A219204 a(n) = [x^n*y^10] Product_{i>=1} (1+x^prime(i)*y).
%p A219204 b:= proc(n, i) option remember; `if`(n=0, [1,0$10], `if`(i<1, [0$11],
%p A219204        zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$10],
%p A219204        b(n-ithprime(i), i-1)[1..10])[]], 0)))
%p A219204     end:
%p A219204 a:= n-> b(n, numtheory[pi](n))[11]:
%p A219204 seq(a(n), n=129..210);
%t A219204 k = 10; 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, 129, 210}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)
%t A219204 Table[Length@Select[IntegerPartitions[k,{10}, Prime@Range@100], #[[1]] > #[[2]] > #[[3]] > #[[4]] > #[[5]] > #[[6]] > #[[7]] > #[[8]] > #[[9]] > #[[10]] &], {k, 129, 210}] (* _Robert Price_, Apr 25 2025 *)
%Y A219204 Column k=10 of A219180.
%K A219204 nonn
%O A219204 129,9
%A A219204 _Alois P. Heinz_, Nov 14 2012