A187757 Number of ways to write n=x+y (x,y>0) with 6x-1, 6x+1, 6y+1 and 6y+5 all prime.

0, 1, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 4, 4, 2, 3, 2, 6, 6, 5, 4, 2, 6, 5, 4, 4, 2, 6, 4, 4, 4, 3, 5, 7, 5, 5, 3, 4, 9, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 6, 6, 3, 7, 7, 6, 6, 4, 6, 6, 5, 6, 4, 7, 6, 7, 2, 3, 7, 7, 7, 5, 3, 5, 5, 7, 8, 5, 8, 8, 4, 5, 4, 10, 10, 6, 6, 2, 9, 6, 9, 7, 1, 8, 4, 5, 7, 3, 9, 5, 3

Offset: 1

Author

Zhi-Wei Sun, Jan 03 2013

Keywords

• nonn
• nice

Comments

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 10^9. It implies that there are infinitely many twin primes and also infinitely many cousin primes, since the interval [m!+2,m!+m] of length m-2 contains no prime for any integer m>1.

Examples

			a(92)=1 since 92=40+52 with 6*40-1, 6*40+1, 6*52+1 and 6*52+5 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A001359, A006512, A023200, A046132, A219157, A218867, A219185, A220455.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]