A259196 Number of partitions of n into six primes.

1, 1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 6, 8, 7, 10, 10, 12, 11, 16, 12, 19, 17, 22, 18, 26, 20, 31, 24, 33, 27, 42, 29, 47, 35, 51, 38, 60, 41, 68, 47, 73, 53, 86, 54, 95, 64, 103, 70, 116, 73, 131, 81, 137, 89, 156, 92, 171, 103, 180, 112, 202, 117, 223, 127, 232

Offset: 12

Author

Doug Bell, Jun 20 2015

Keywords

• nonn
• easy

Examples

			a(17) = 3 because there are 3 partitions of 17 into six primes: [2,2,2,2,2,7], [2,2,2,3,3,5] and [2,3,3,3,3,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 12..10000
• Index entries for sequences related to partitions

Crossrefs

Column k=6 of A117278.

Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, this sequence, A259197, A259198, A259200, A259201.

Cf. A000040.

Programs

Mathematica

Table[Count[IntegerPartitions[n,{6}],?(AllTrue[#,PrimeQ]&)],{n,12,80}] (* _Harvey P. Dale, Jul 27 2024 *)

Formula

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

a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019