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A019334 Primes with primitive root 3.

%I A019334 #38 Feb 16 2025 08:32:33
%S A019334 2,5,7,17,19,29,31,43,53,79,89,101,113,127,137,139,149,163,173,197,
%T A019334 199,211,223,233,257,269,281,283,293,317,331,353,379,389,401,449,461,
%U A019334 463,487,509,521,557,569,571,593,607,617,631,641,653,677,691,701,739,751,773,797
%N A019334 Primes with primitive root 3.
%C A019334 To allow primes less than the specified primitive root m (here, 3) 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 A019334 From _Jianing Song_, Apr 27 2019: (Start)
%C A019334 All terms except the first are congruent to 5 or 7 modulo 12. If we define
%C A019334   Pi(N,b) = # {p prime, p <= N, p == b (mod 12)};
%C A019334      Q(N) = # {p prime, 2 < p <= N, p in this sequence},
%C A019334 then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,5) + Pi(N,7)), where C = A005596 is Artin's constant.
%C A019334 If we further define
%C A019334    Q(N,b) = # {p prime, p <= N, p == b (mod 12), p in this sequence},
%C A019334 then we have:
%C A019334    Q(N,5) ~ (3/5)*Q(N) ~ (12/5)*C*Pi(N,5);
%C A019334    Q(N,7) ~ (2/5)*Q(N) ~ ( 8/5)*C*Pi(N,7).
%C A019334 For example, for the first 1000 terms except for a(1) = 2, there are 593 terms == 5 (mod 12) and 406 terms == 7 (mod 12). (End)
%H A019334 Vincenzo Librandi, <a href="/A019334/b019334.txt">Table of n, a(n) for n = 1..1000</a>
%H A019334 J. Conde, M. Miller, J. M. Miret, K. Saurav, <a href="https://www.researchgate.net/publication/264971121_On_the_nonexistence_of_almost_Moore_digraphs_of_degree_five">On the Nonexistence of Almost Moore Digraphs of Degree Four and Five</a>, International Conference on Mathematical Computer Engineering (ICMCE-13), pp. 2-7, At VIT University, Chennai, Volume: I, 2013.
%H A019334 J. Conde, M. Miller, J. M. Miret, K. Saurav, <a href="https://doi.org/10.1007/s11786-015-0219-z">On the Nonexistence of Almost Moore Digraphs of Degree Four and Five</a>, Mathematics in Computer Science, June 2015, Volume 9, Issue 2, pp 145-149.
%H A019334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>
%H A019334 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>
%H A019334 <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>
%t A019334 pr=3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
%o A019334 (PARI) isok(p) = isprime(p) && (p!=3) && (znorder(Mod(3, p))+1 == p); \\ _Michel Marcus_, May 12 2019
%Y A019334 Cf. A005596, A001122 (primitive root 2).
%Y A019334 Cf. A019335-A019421.
%K A019334 nonn
%O A019334 1,1
%A A019334 _David W. Wilson_