

A019335


Primes with primitive root 5.


9



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



