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

1,1

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

Mathematica

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

Prog

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

Crossrefs

Cf. A005596, A001122 (primitive root 2).

Cf. A019335-A019421.

Sequence in context: A174281 A301916 A038875 * A045356 A158526 A146364

Adjacent sequences: A019331 A019332 A019333 * A019335 A019336 A019337

Keyword

nonn

Author

David W. Wilson

Status

approved