A219203 Number of partitions of n into 9 distinct primes.
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, 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, 97, 9, 91, 18, 96, 25, 130, 16, 131, 30, 134, 35, 177, 25, 182
Offset: 100
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 100..10000
Crossrefs
Column k=9 of A219180.
Programs
-
Maple
b:= proc(n, i) option remember; `if`(n=0, [1,0$9], `if`(i<1, [0$10], zip((x, y)->x+y, b(n, i-1), [0, `if`(ithprime(i)>n, [0$9], b(n-ithprime(i), i-1)[1..9])[]], 0))) end: a:= n-> b(n, numtheory[pi](n))[10]: seq(a(n), n=100..180); -
Mathematica
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 *) Table[Length[Select[IntegerPartitions[n,{9}],AllTrue[#,PrimeQ]&&Length[Union[#]] == 9&]],{n,100,180}] (* Harvey P. Dale, Mar 09 2023 *)
Formula
G.f.: Sum_{0
a(n) = [x^n*y^9] Product_{i>=1} (1+x^prime(i)*y).