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

Author

Doug Bell, Jun 20 2015

Keywords

• nonn
• easy

Examples

			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].
		

Links

• Robert Israel, Table of n, a(n) for n = 18..10000
• Index entries for sequences related to partitions

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.

Programs

Magma

[#RestrictedPartitions(k,9,Set(PrimesUpTo(1000))):k in [18..70]] ; // Marius A. Burtea, Jul 13 2019

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 *)

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

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)} c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q), where c = A010051. - Wesley Ivan Hurt, Jul 13 2019