A323617 Primes p such that 3 is a primitive root modulo p while 243 is not.

31, 101, 211, 281, 331, 401, 461, 521, 571, 631, 641, 691, 701, 751, 811, 821, 881, 941, 1061, 1231, 1291, 1301, 1361, 1481, 1601, 1721, 1831, 1901, 1951, 2011, 2081, 2141, 2311, 2371, 2381, 2731, 2741, 2801, 2861, 3041, 3271, 3331, 3391, 3461, 3571, 3581, 3701, 3761, 3821, 3931

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 5).

According to Artin's conjecture, the number of terms <= N is roughly ((4/19)*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

• Amiram Eldar, 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), this sequence (q=5), A323628 (q=7).

Programs

PARI

forprime(p=5, 4000, if(znorder(Mod(3, p))==(p-1) && p%5==1, print1(p, ", ")))