A259200 Number of partitions of n into nine primes.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 7, 9, 10, 11, 12, 16, 16, 20, 21, 24, 26, 33, 31, 39, 39, 47, 46, 59, 53, 69, 65, 80, 77, 98, 85, 114, 104, 131, 118, 154, 133, 179, 155, 200, 177, 236, 196, 268, 227, 300, 256

Offset

18,4

Links

Robert Israel, Table of n, a(n) for n = 18..10000

Index entries for sequences related to partitions

Formula

a(n) = [x^n y^9] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} A010051(q) * A010051(p) * A010051(o) * A010051(m) * A010051(l) * A010051(k) * A010051(j) * A010051(i) * A010051(n-i-j-k-l-m-o-p-q). - Wesley Ivan Hurt, Jul 13 2019

Example

a(23) = 3 because there are 3 partitions of 23 into nine primes: [2,2,2,2,2,2,2,2,7], [2,2,2,2,2,2,3,3,5] and [2,2,2,2,3,3,3,3,3].

Maple

N:= 100: # to get a(0) to a(N)
Primes:= select(isprime, [$1..N]):
np:= nops(Primes):
for j from 0 to np do g[0, j]:= 1 od:
for n from 1 to 9 do
  g[n, 0]:= 0:
  for j from 1 to np do
     g[n, j]:= convert(series(add(g[k, j-1]
          *x^((n-k)*Primes[j]), k=0..n), x, N+1), polynom)
  od
od:
seq(coeff(g[9, np], x, i), i=18..N) # Robert Israel, Jun 21 2015

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[#]==9&&AllTrue[ #, PrimeQ]&]], {n, 18, 70}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 31 2016 *)

Prog

(PARI) a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [9, 9]); nb; } \\ Michel Marcus, Jun 21 2015
(Magma) [#RestrictedPartitions(k, 9, Set(PrimesUpTo(1000))):k in [18..70]] ; // Marius A. Burtea, Jul 13 2019

Crossrefs

Column k=9 of A117278.

Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, this sequence, A259201.

Cf. A000040.

Sequence in context: A029042 A320470 A320382 * A153155 A225085 A134310

Adjacent sequences: A259197 A259198 A259199 * A259201 A259202 A259203

Keyword

nonn,easy

Author

Doug Bell, Jun 20 2015

Status

approved