A323590 Primes p such that 2 is a primitive root modulo p while 8192 is not.

53, 131, 443, 547, 677, 859, 1171, 1301, 1483, 2029, 2237, 2549, 2861, 2939, 3797, 4603, 5227, 5851, 6397, 6709, 6917, 7229, 7307, 7411, 7541, 7853, 8243, 8269, 8867, 8971, 9283, 9491, 9803, 9907, 10037, 10141, 10427, 10973, 11779, 11909, 11987, 12611, 12637, 12923

Offset: 1

Author

Jianing Song, Aug 30 2019

Keywords

• nonn

Comments

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 13).

According to Artin's conjecture, the number of terms <= N is roughly ((12/155)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Artin's constant
• Wikipedia, Artin's conjecture on primitive roots

Crossrefs

Cf. A001122, A005596, A000720.

Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), A323576 (q=7), A323577 (q=11), this sequence (q=13).

Programs

Maple

filter:= proc(p) isprime(p) and numtheory:-order(2,p) = p-1 end proc:
select(filter, [seq(i, i = 1 .. 13000, 26)]); # Robert Israel, Dec 20 2023

PARI

forprime(p=3, 13000, if(znorder(Mod(2, p))==(p-1) && p%13==1, print1(p, ", ")))