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

1,8

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

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A201208 A006513 A105224 * A237112 A238013 A303940

Adjacent sequences: A261624 A261625 A261626 * A261628 A261629 A261630

Keyword

nonn

Author

Zhi-Wei Sun, Aug 27 2015

Status

approved