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A219203 Number of partitions of n into 9 distinct primes.

%I A219203 #15 Mar 09 2023 14:25:11
%S A219203 1,0,0,0,0,0,1,0,1,0,1,0,2,0,2,0,1,0,5,0,4,0,2,0,9,0,7,1,6,1,13,0,10,
%T A219203 0,12,2,20,0,19,2,20,3,31,1,30,4,28,5,49,3,45,7,43,9,69,7,63,10,66,16,
%U A219203 97,9,91,18,96,25,130,16,131,30,134,35,177,25,182
%N A219203 Number of partitions of n into 9 distinct primes.
%H A219203 Alois P. Heinz, <a href="/A219203/b219203.txt">Table of n, a(n) for n = 100..10000</a>
%F A219203 G.f.: Sum_{0<i_1<i_2<...<i_9} x^(Sum_{j=1..9} prime(i_j)).
%F A219203 a(n) = [x^n*y^9] Product_{i>=1} (1+x^prime(i)*y).
%p A219203 b:= proc(n, i) option remember; `if`(n=0, [1,0$9], `if`(i<1, [0$10],
%p A219203        zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$9],
%p A219203        b(n-ithprime(i), i-1)[1..9])[]], 0)))
%p A219203     end:
%p A219203 a:= n-> b(n, numtheory[pi](n))[10]:
%p A219203 seq(a(n), n=100..180);
%t A219203 k = 9; 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, 100, 180}] (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)
%t A219203 Table[Length[Select[IntegerPartitions[n,{9}],AllTrue[#,PrimeQ]&&Length[Union[#]] == 9&]],{n,100,180}] (* _Harvey P. Dale_, Mar 09 2023 *)
%Y A219203 Column k=9 of A219180.
%K A219203 nonn
%O A219203 100,13
%A A219203 _Alois P. Heinz_, Nov 14 2012