A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 03 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>4.

This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.

Examples

			a(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+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. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==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}]