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A019335 Primes with primitive root 5.

%I A019335 #34 Feb 16 2025 08:32:33
%S A019335 2,3,7,17,23,37,43,47,53,73,83,97,103,107,113,137,157,167,173,193,197,
%T A019335 223,227,233,257,263,277,283,293,307,317,347,353,373,383,397,433,443,
%U A019335 463,467,503,523,547,557,563,577,587,593,607,613,617,647,653,673,677,683,727
%N A019335 Primes with primitive root 5.
%C A019335 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
%C A019335 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
%C A019335 From _Jianing Song_, Apr 27 2019: (Start)
%C A019335 If we define
%C A019335   Pi(N,b) = # {p prime, p <= N, p == b (mod 5)};
%C A019335      Q(N) = # {p prime, p <= N, p in this sequence},
%C A019335 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.
%C A019335 Conjecture: if we further define
%C A019335    Q(N,b) = # {p prime, p <= N, p == b (mod 5), p in this sequence},
%C A019335 then we have:
%C A019335    Q(N,2) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,2);
%C A019335    Q(N,3) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,3). (End)
%H A019335 Vincenzo Librandi, <a href="/A019335/b019335.txt">Table of n, a(n) for n = 1..1000</a>
%H A019335 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>
%H A019335 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>
%H A019335 <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>
%t A019335 pr=5; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
%o A019335 (PARI) isok(p) = isprime(p) && (p != 5) && (znorder(Mod(5, p)) == p-1); \\ _Michel Marcus_, Apr 27 2019
%Y A019335 Cf. A019334-A019421.
%K A019335 nonn
%O A019335 1,1
%A A019335 _David W. Wilson_