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

0,3

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.

Links

Table of n, a(n) for n=0..73.

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.

Example

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.

Sequence in context: A000091 A155123 A125938 * A158851 A151930 A356907

Adjacent sequences: A215458 A215459 A215460 * A215462 A215463 A215464

Keyword

nonn

Author

Peter A. Lawrence, Aug 11 2012

Status

approved