A220091 Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 1, 2, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 1, 4, 2, 1, 3, 1, 1, 3, 4, 3, 3, 2, 1, 1, 3, 3, 1, 2, 2, 4, 4, 5, 3, 1, 1, 3, 2, 3, 3, 2, 2, 4, 2, 3, 3, 0, 1, 5, 2, 2, 3, 1, 0, 2, 3

Offset: 1

Author

Zhi-Wei Sun, Dec 04 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all even n>=8070 and odd n>=18680.

This conjecture unifies the twin prime conjecture, Goldbach's conjecture and Lemoine's conjecture. It has been verified for n up to 10^7.

Zhi-Wei Sun also made the following conjecture: Any integer n>=6782 can be written as p+q+(n mod 2)q with p>q and p, q, q-6, q+6 all prime, and any integer n>=4410 can be written as p+q+(n mod 2)q with p>q and p, q, 2q-3, 2q+3 all prime, and any integer n>=16140 can be written as p+q+(n mod 2)q with p>q and p, q, 3q-2, 3q+2 all prime.

Examples

			a(31)=1 since 31=17+2*7 with 6*7-1 and 6*7+1 twin primes.
a(32)=1 since 32=29+3 with 6*3-1 and 6*3+1 twin primes.
		

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, A219185, A219157, A218867, A219055, A219966.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[6Prime[k]-1]==True&&PrimeQ[6Prime[k]+1]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,100}]