cp's OEIS Frontend

A261628 Least prime p such that n-(p*n'-1) and n+(p*n'-1) are both prime where n' = (3+(-1)^n)/2, or 0 if no such prime p exists.

%I A261628 #25 Aug 04 2025 05:36:00
%S A261628 0,0,0,0,3,0,5,2,3,2,7,3,7,2,3,2,7,3,13,2,3,5,7,3,7,2,5,5,13,7,13,5,5,
%T A261628 2,7,3,7,5,3,2,13,3,31,2,3,17,7,3,13,2,11,5,7,7,13,2,5,11,13,7,19,5,5,
%U A261628 2,7,3,7,11,3,2,13,13,7,17,5,2,7,3,19,5
%N A261628 Least prime p such that n-(p*n'-1) and n+(p*n'-1) are both prime where n' = (3+(-1)^n)/2, or 0 if no such prime p exists.
%C A261628 Conjecture: 0 < a(n) < sqrt(2*n)*log(5*n) for all n > 6.
%C A261628 See also A261627.
%C A261628 Verified up to 10^9. - _Mauro Fiorentini_, Jul 05 2023
%C A261628 Conjecture verified for n < 1.2 * 10^12. Also, the 5 inside the log function can probably be replaced by 4.26. - _Jud McCranie_, Aug 26 2023
%H A261628 Zhi-Wei Sun, <a href="/A261628/b261628.txt">Table of n, a(n) for n = 1..10000</a>
%H A261628 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 A261628 a(43) = 31 since 31, 43-(31-1) = 13 and 43+(31-1) = 73 are all prime.
%e A261628 a(72) = 13 since 13, 72-(2*13-1) = 47 and 72+(2*13-1) = 97 are all prime.
%t A261628 Do[Do[If[PrimeQ[n-(3+(-1)^n)/2*Prime[k]+1]&&PrimeQ[n+(3+(-1)^n)/2*Prime[k]-1],Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[2n/(3+(-1)^n)]}];Print[n," ",0];Label[aa];Continue,{n,1,80}]
%Y A261628 Cf. A000040, A002372, A002375, A046927, A261627.
%K A261628 nonn
%O A261628 1,5
%A A261628 _Zhi-Wei Sun_, Aug 27 2015