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A019335
Primes with primitive root 5.
11
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)
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
KEYWORD
nonn
STATUS
approved