

A019335


Primes with primitive root 5.


10



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
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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 p1". This comment applies to all of A019334A019421.  N. J. A. Sloane, Dec 02 2019
Appears to be the numbers k such that the sequence 5^n mod k has period length k1. 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)) == p1); \\ Michel Marcus, Apr 27 2019


CROSSREFS

Cf. A019334A019421.
Sequence in context: A045333 A040141 A235627 * A113425 A289379 A245590
Adjacent sequences: A019332 A019333 A019334 * A019336 A019337 A019338


KEYWORD

nonn


AUTHOR

David W. Wilson


STATUS

approved



