A019335 Primes with primitive root 5.

2, 3, 7, 17, 23, 37, 43, 47, 53, 73, 83, 97, 103, 107, 113, 137, 157, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 317, 347, 353, 373, 383, 397, 433, 443, 463, 467, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 617, 647, 653, 673, 677, 683, 727

Offset

1,1

Comments

To allow primes less than the specified primitive root m (here, 5) 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

Appears to be the numbers k such that the sequence 5^n mod k has period length k-1. All terms are congruent to 2 or 3 mod 5. - Gary Detlefs, May 21 2014

From Jianing Song, Apr 27 2019: (Start)

If we define

Pi(N,b) = # {p prime, p <= N, p == b (mod 5)};

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

then by Artin's conjecture, Q(N) ~ (20/19)*C*N/log(N) ~ (40/19)*C*(Pi(N,2) + Pi(N,3)), where C = A005596 is Artin's constant.

Conjecture: if we further define

Q(N,b) = # {p prime, p <= N, p == b (mod 5), p in this sequence},

then we have:

Q(N,2) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,2);

Q(N,3) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,3). (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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=5; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

Prog

(PARI) isok(p) = isprime(p) && (p != 5) && (znorder(Mod(5, p)) == p-1); \\ Michel Marcus, Apr 27 2019

Crossrefs

Cf. A019334-A019421.

Sequence in context: A045333 A040141 A235627 * A378135 A113425 A289379

Adjacent sequences: A019332 A019333 A019334 * A019336 A019337 A019338

Keyword

nonn

Author

David W. Wilson

Status

approved