A051899 Smallest prime p such that (p-1)/(2*n) is also a prime.

5, 13, 13, 17, 31, 37, 29, 113, 37, 41, 67, 73, 53, 197, 61, 97, 103, 73, 191, 281, 127, 89, 139, 97, 101, 157, 109, 113, 1103, 181, 311, 193, 199, 137, 211, 937, 149, 229, 157, 241, 1559, 421, 173, 617, 181, 277, 283, 193, 197, 701, 307, 313, 743, 541, 331

Offset: 1

Author

Labos Elemer, Dec 16 1999

Keywords

• nonn
• easy

Examples

			a(2) = 13 because (13 - 1)/4 = 3 = A051686(2).
a(3) = 13 as well, because (13 - 1)/6 = 2 = A051686(3).
a(8) = 113 because (113 - 1)/16 = 7 is a prime.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005385, A051686.

Programs

Mathematica

a[n_] := Module[{p = 2}, While[!PrimeQ[2*n*p + 1], p = NextPrime[p]]; 2*n*p + 1]; Array[a, 55] (* Amiram Eldar, Feb 25 2025 *)

PARI

isp(q) = (denominator(q)==1) && isprime(q);
a(n) = {my(p = 2); while (!isp((p-1)/(2*n)), p = nextprime(p+1)); p;} \\ Michel Marcus, May 29 2018

PARI

a(n) = forprime(p = 2, oo, q = 2 * p * n + 1; if(isprime(q), return(q))) \\ David A. Corneth, May 29 2018

Formula

(a(n)-1)/2n = A051686(n), the smallest 2k-Germain primes.