A215461 Number of decompositions of 2n into ordered sums of one prime and one nonprime.

0, 0, 2, 2, 2, 0, 4, 4, 2, 4, 6, 4, 4, 6, 8, 6, 12, 6, 4, 16, 10, 8, 14, 12, 8, 12, 16, 10, 18, 16, 8, 24, 14, 10, 28, 16, 14, 22, 20, 12, 26, 24, 12, 26, 28, 10, 30, 28, 18, 36, 24, 18, 32, 30, 22, 32, 28, 18, 34, 36, 10, 44, 38, 18, 48, 32, 26, 40, 42, 32, 38, 36, 22, 44

Offset: 0

Author

Peter A. Lawrence, Aug 11 2012

Keywords

• nonn

Comments

A002372(n) + a(n) + A215462(n) = n.

Note: a(n) always even.

Conjecture: a(n) is never zero for n > 5, verified to 10^9.

Goldbach conjecture: a(n) + A215462(n) < n for all n > 2.

Examples

			n=15, 2*n=30, 2*n = { 3+27, 5+25, 29+1;  1+29, 25+5, 27+3 }, a(15) = 6
n=18, 2*n=36, 2*n = { 3+33, 11+25;  11+25, 33+3 }, a(18) = 4
		

Crossrefs

Cf. A002372, A215462.

Formula

a(n) = convolve(p,c) + convolve(c,p) = 2*convolve(p,c) where p(n) = 1 if 2n+1 is prime and 0 otherwise, and c(n) = 1 if 2n+1 is nonprime and 0 otherwise.