A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime.

0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 0, 4, 2, 4, 3, 2, 2, 3, 3, 5, 3, 3, 3, 4, 4, 3, 2, 4, 3, 5, 3, 4, 3, 5, 5, 6, 3, 4, 3, 5, 5, 5, 6, 5, 4, 5, 5, 6, 7, 5, 4, 5, 4, 7, 6, 4, 4, 4, 5, 6, 6, 5, 6, 7, 4, 5, 2, 4, 7, 5, 7, 4, 5, 6, 7, 7, 7, 5, 6, 4, 7, 4, 7, 7, 6, 5, 3, 5, 8, 7, 7, 5, 5, 6, 4, 5, 4, 5, 8, 7

Offset: 1

Author

Zhi-Wei Sun, Dec 21 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>11.

This implies the twin prime conjecture, and it has been verified for n up to 10^9.

Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.

Examples

			a(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 all 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. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]