A219842 Number of ways to write n as x+y (0

0, 1, 1, 1, 1, 3, 1, 1, 4, 2, 2, 4, 2, 2, 7, 4, 1, 3, 4, 5, 5, 4, 2, 8, 2, 7, 4, 2, 8, 11, 5, 3, 8, 7, 5, 14, 7, 5, 10, 8, 7, 8, 4, 8, 9, 5, 4, 11, 6, 11, 14, 5, 3, 19, 12, 7, 11, 6, 9, 12, 13, 8, 9, 10, 12, 16, 5, 6, 22, 8, 11, 11, 5, 10, 26, 15, 5, 11, 15, 10

Offset: 1

Author

Zhi-Wei Sun, Nov 29 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>1. Moreover, any integer n>357 can be written as x+y (x>0, y>0) with 2x*y+1 and 2x*y-1 twin primes.

This conjecture has been verified for n up to 10^8, and it implies the twin prime conjecture.

Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y, where x and y are positive integers with 2x*y+m and 2x*y-m both prime.

For example, when m=3,5,7,9,11 it suffices to require that n is greater than 5090, 222, 1785, 548, 603 respectively.

Examples

			a(10)=2 since 10=1+9=3+7 with 2*1*9+1=19 and 2*3*7+1=43 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A091182, A219838, A219782.

Programs

Mathematica

a[n_] := a[n] = Sum[If[PrimeQ[2k(n-k)+1] == True, 1, 0], {k, n/2}]; Do[Print[n," ", a[n]], {n, 100}]