A259198 Number of partitions of n into eight primes.

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 9, 12, 14, 16, 16, 21, 19, 26, 26, 31, 30, 39, 34, 46, 43, 53, 48, 65, 56, 77, 66, 85, 77, 104, 84, 118, 99, 133, 112, 155, 123, 177, 143, 196, 162, 227, 174, 256, 200, 282, 220, 318, 241, 360, 270, 389, 300, 442, 322

Offset: 16

Author

Doug Bell, Jun 20 2015

Keywords

• nonn
• easy

Examples

			a(20) = 2 because there are 2 partitions of 20 into eight primes: [2,2,2,2,2,2,3,5] and [2,2,2,2,3,3,3,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 16..10000
• Index entries for sequences related to partitions

Crossrefs

Column k=8 of A117278.

Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, A259196, A259197, this sequence, A259200, A259201.

Cf. A000040.

Programs

Mathematica

Table[Length@IntegerPartitions[n, {8}, Prime@Range@100], {n, 16, 100}] (* Robert Price, Apr 25 2025 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(l) * A010051(m) * A010051(o) * A010051(p) * A010051(n-i-j-k-l-m-o-p). - Wesley Ivan Hurt, Apr 17 2019

a(n) = [x^n y^8] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

a(n) = A326455(n)/n for n > 0. - Wesley Ivan Hurt, Jul 06 2019