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.

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, 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, 2, 7, 3, 7, 11, 3, 2, 13, 13, 7, 17, 5, 2, 7, 3, 19, 5

Offset: 1

Author

Zhi-Wei Sun, Aug 27 2015

Keywords

• nonn

Comments

Conjecture: 0 < a(n) < sqrt(2*n)*log(5*n) for all n > 6.

See also A261627.

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

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

Examples

			a(43) = 31 since 31, 43-(31-1) = 13 and 43+(31-1) = 73 are all prime.
a(72) = 13 since 13, 72-(2*13-1) = 47 and 72+(2*13-1) = 97 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, A261627.

Programs

Mathematica

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}]