A307857 Number of partitions of n into 1, 2 or 3 nonprime parts.

1, 1, 1, 1, 1, 2, 1, 3, 3, 4, 3, 5, 4, 6, 5, 9, 7, 10, 8, 12, 10, 15, 11, 18, 15, 20, 17, 24, 19, 28, 22, 30, 26, 36, 29, 41, 34, 42, 37, 51, 41, 55, 47, 59, 53, 66, 54, 73, 63, 78, 70, 85, 72, 94, 81, 99, 89, 108, 92, 118, 102, 121, 110, 135, 117, 143, 126

Offset: 1

Author

Wesley Ivan Hurt, May 01 2019

Keywords

• nonn
• easy

Examples

			a(9) = 3, because 9 can be written as the sum of nonprimes with at most 3 parts in three ways: 9 = 8+1 = 4+4+1.
a(10) = 4, because 10 can be written as the sum of nonprimes with at most 3 parts in four ways: 10 = 9+1 = 6+4 = 8+1+1.
a(11) = 3, because 11 can be written as the sum of nonprimes with at most 3 parts in three ways: 10+1 = 9+1+1 = 6+4+1.
a(12) = 5, because 12 can be written as the sum of nonprimes with at most 3 parts in five ways: 12 = 8+4 = 6+6 = 10+1+1 = 4+4+4.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A005171, A018252, A071335.

Formula

a(n) = c(n) + ( Sum_{i=1..floor(n/2)} c(i) * c(n-i) ) + ( Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) ), where c = A005171.