A019334 Primes with primitive root 3.

2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, 269, 281, 283, 293, 317, 331, 353, 379, 389, 401, 449, 461, 463, 487, 509, 521, 557, 569, 571, 593, 607, 617, 631, 641, 653, 677, 691, 701, 739, 751, 773, 797

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

To allow primes less than the specified primitive root m (here, 3) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019

From Jianing Song, Apr 27 2019: (Start)

All terms except the first are congruent to 5 or 7 modulo 12. If we define

Pi(N,b) = # {p prime, p <= N, p == b (mod 12)};

Q(N) = # {p prime, 2 < p <= N, p in this sequence},

then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,5) + Pi(N,7)), where C = A005596 is Artin's constant.

If we further define

Q(N,b) = # {p prime, p <= N, p == b (mod 12), p in this sequence},

then we have:

Q(N,5) ~ (3/5)*Q(N) ~ (12/5)*C*Pi(N,5);

Q(N,7) ~ (2/5)*Q(N) ~ ( 8/5)*C*Pi(N,7).

For example, for the first 1000 terms except for a(1) = 2, there are 593 terms == 5 (mod 12) and 406 terms == 7 (mod 12). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, International Conference on Mathematical Computer Engineering (ICMCE-13), pp. 2-7, At VIT University, Chennai, Volume: I, 2013.
• J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, June 2015, Volume 9, Issue 2, pp 145-149.
• Eric Weisstein's World of Mathematics, Artin's constant
• Wikipedia, Artin's conjecture on primitive roots
• Index entries for primes by primitive root

Crossrefs

Cf. A005596, A001122 (primitive root 2).

Cf. A019335-A019421.

Programs

Mathematica

pr=3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

PARI

isok(p) = isprime(p) && (p!=3) && (znorder(Mod(3, p))+1 == p); \\ Michel Marcus, May 12 2019