A261627 Number of primes p such that n-(p*n'-1) and n+(p*n'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

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

Offset: 1

Author

Zhi-Wei Sun, Aug 27 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 6, and a(n) = 1 only for n = 5, 7, 10, 11, 12, 19, 22, 30, 34, 44, 46, 72, 142.

This is stronger than Goldbach's conjecture (A002375) and Lemoine's conjecture (A046927).

I have verified the conjecture for n up to 10^8.

Verified for n up to 10^9. - Mauro Fiorentini, Jul 05 2023

Conjecture verified for n < 1.2 * 10^12. - Jud McCranie, Aug 26 2023

Examples

			a(19) = 1 since 13, 19-(13-1) = 7 and 19+(13-1) = 31 are all prime.
a(142) = 1 since 41, 142-(2*41-1) = 61 and 142+(2*41-1) = 223 are 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 [math.NT], 2012-2015.

Crossrefs

Cf. A000040, A002372, A002375, A046927, A219055, A237284, A261628.

Programs

Mathematica

Do[r=0;Do[If[PrimeQ[n-(3+(-1)^n)/2*Prime[k]+1]&&PrimeQ[n+(3+(-1)^n)/2*Prime[k]-1],r=r+1],{k,1,PrimePi[2n/(3+(-1)^n)]}];Print[n," ",r];Continue,{n,1,80}]