A323628 Primes p such that 3 is a primitive root modulo p while 2187 is not.

29, 43, 113, 127, 197, 211, 281, 379, 449, 463, 617, 631, 701, 953, 1373, 1709, 1723, 2129, 2143, 2213, 2311, 2381, 2549, 2633, 2647, 2731, 2801, 2969, 3137, 3389, 3557, 3571, 3823, 4159, 4229, 4243, 4327, 4397, 4481, 4649, 4663, 4817, 4831, 4999, 5237, 5419

Offset: 1

Author

Jianing Song, Aug 30 2019

Keywords

• nonn

Comments

Primes p such that 3 is a primitive root modulo p (i.e., p is in A019334) and that p == 1 (mod 7).

According to Artin's conjecture, the number of terms <= N is roughly ((6/41)*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. A019334, A005596, A000720.

Primes p such that 3 is a primitive root modulo p and that p == 1 (mod q): A323594 (q=3), A323617 (q=5), this sequence (q=7).

Programs

Maple

select(p -> isprime(p) and numtheory:-order(3,p)=p-1, [seq(i,i=1..10000,7)]); # Robert Israel, Sep 01 2019

PARI

forprime(p=5, 5500, if(znorder(Mod(3, p))==(p-1) && p%7==1, print1(p, ", ")))