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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 (list; graph; refs; listen; history; text; internal format)
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
Eric Weisstein's World of Mathematics, Artin's constant
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
Sequence in context: A045333 A040141 A235627 * A113425 A289379 A245590
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 29 05:28 EDT 2024. Contains 371264 sequences. (Running on oeis4.)