A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Dec 22 2012

Keywords

• nonn
• nice

Comments

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

This implies that there are infinitely many twin primes and also infinitely many sexy primes. It has been verified for n up to 10^9. See also A199800 for a weaker version of this conjecture.

Zhi-Wei Sun also conjectured that any integer n>6 not equal to 319 can be written as p+k with p, p+6, 3k-2+(n mod 2) and 3k+2-(n mod 2) all prime.

Examples

			a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A001359, A006512, A023201, A199800, A219055, A219864, A219923, A002375.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[6(n-Prime[k])-1]==True&&PrimeQ[6(n-Prime[k])+1]==True,1,0],{k,1,PrimePi[n]}]
Do[Print[n," ",a[n]],{n,1,100}]
Table[Count[Table[{n-i,i},{i,n-1}],?(AllTrue[{#[[1]],#[[1]]+6,6#[[2]]-1,6#[[2]]+1},PrimeQ]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, May 19 2015 *)

PARI

a(n)=my(s,p=2,q=3); forprime(r=5,n+5, if(r-p==6 && isprime(6*n-6*p-1) && isprime(6*n-6*p+1), s++); if(r-q==6 && isprime(6*n-6*q-1) && isprime(6*n-6*q+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Jul 31 2016