A219966 Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Dec 02 2012

Keywords

• nonn

Comments

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

This conjecture is stronger than Goldbach's conjecture and Lemoine's conjecture. It can be further strengthened; see A219055 and the comments there.

Examples

			a(19)=1 since 19=5+2*7 with 5, 7, 7+6 all prime.
a(20)=1 since 20=13+7 with 13, 7, 7+6 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. A219055, A023201, A002375, A046927.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[n/2]}]
Do[Print[n," ",a[n]],{n,1,10000}]