A219157 Number of prime pairs {p,q} with p>q and p-2,q+2 also prime such that p+(1+mod(-n,6))q=n if n is not congruent to 2 mod 6, and p-q=n and q

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 0, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 2, 1, 1, 0, 2, 2, 1, 2, 1, 2, 1, 0, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 0, 1, 3, 1, 0

Offset: 1

Author

Zhi-Wei Sun, Nov 12 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>30000 with n different from 38451, 46441, 50671, 62371.

This conjecture is stronger than the twin prime conjecture. It is similar to the conjecture associated with A219055 about sexy prime pairs.

Examples

			a(16)=1 since 16=7+3*3 with 7-2 and 3+2 prime. a(26)=1 since 26=31-5 with 31-2 and 5+2 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A001359, A006512, A002375, A046927, A219055, A218754, A218825, A219052.

Programs

Mathematica

c[n_]:=c[n]=If[Mod[n-2,6]==0,1,-1-Mod[-n,6]]
d[n_]:=d[n]=2+If[Mod[n-2,6]>0,Mod[-n,6],0]
a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[n+c[n]Prime[k]]==True&&PrimeQ[n+c[n]Prime[k]-2]==True,1,0],
{k,1,PrimePi[(n-1)/d[n]]}]
Do[Print[n," ",a[n]],{n,1,100000}]