cp's OEIS Frontend

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.

%I A261627 #32 Sep 07 2023 16:11:14
%S A261627 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,
%T A261627 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,
%U A261627 5,4,6,1,4,2,4,3,5,4,3,4
%N 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.
%C A261627 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.
%C A261627 This is stronger than Goldbach's conjecture (A002375) and Lemoine's conjecture (A046927).
%C A261627 I have verified the conjecture for n up to 10^8.
%C A261627 Verified for n up to 10^9. - _Mauro Fiorentini_, Jul 05 2023
%C A261627 Conjecture verified for n < 1.2 * 10^12. - _Jud McCranie_, Aug 26 2023
%H A261627 Zhi-Wei Sun, <a href="/A261627/b261627.txt">Table of n, a(n) for n = 1..10000</a>
%H A261627 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2015.
%e A261627 a(19) = 1 since 13, 19-(13-1) = 7 and 19+(13-1) = 31 are all prime.
%e A261627 a(142) = 1 since 41, 142-(2*41-1) = 61 and 142+(2*41-1) = 223 are all prime.
%t A261627 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}]
%Y A261627 Cf. A000040, A002372, A002375, A046927, A219055, A237284, A261628.
%K A261627 nonn
%O A261627 1,8
%A A261627 _Zhi-Wei Sun_, Aug 27 2015