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A105887 Primes for which -15 is a primitive root.

%I A105887 #40 Feb 16 2025 08:32:57
%S A105887 2,11,13,29,37,41,43,59,71,73,89,97,101,103,127,131,149,157,163,179,
%T A105887 191,193,239,251,269,281,307,313,337,359,373,389,401,419,431,433,449,
%U A105887 457,461,479,487,491,509,521,523,547,569,577,599,607,613,641,701,719,727,733,757
%N A105887 Primes for which -15 is a primitive root.
%C A105887 From _Jianing Song_, Jan 27 2019: (Start)
%C A105887 All terms except the first are congruent to 7, 11, 13 or 14 modulo 15. If we define
%C A105887   Pi(N,b) = # {p prime, p <= N, p == b (mod 15)};
%C A105887      Q(N) = # {p prime, 2 < p <= N, p in this sequence},
%C A105887 then by Artin's conjecture, Q(N) ~ (94/95)*C*N/log(N) ~ (188/95)*C*(Pi(N,7) + Pi(N,11) + Pi(N,13) + Pi(N,14)), where C = A005596 is Artin's constant.
%C A105887 Conjecture: if we further define
%C A105887    Q(N,b) = # {p prime, p <= N, p == b (mod 15), p in this sequence},
%C A105887 then we have:
%C A105887    Q(N,7) ~ (10/47)*Q(N) ~ ( 80/95)*C*Pi(N,7);
%C A105887   Q(N,11) ~ (12/47)*Q(N) ~ ( 96/95)*C*Pi(N,11);
%C A105887   Q(N,13) ~ (10/47)*Q(N) ~ ( 80/95)*C*Pi(N,13);
%C A105887   Q(N,14) ~ (15/47)*Q(N) ~ (120/95)*C*Pi(N,14).
%C A105887 Numeric verification up tp N = 10^8:
%C A105887                |  Q(N,7) | Q(N,11) | Q(N,13) | Q(N,14) |    Q(N)
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887       N = 10^3 |      14 |      18 |      13 |      19 |      64
%C A105887    Q(N,*)/Q(N) | 0.21875 | 0.28125 | 0.20313 | 0.29688 | 1.00000
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887       N = 10^4 |      97 |     115 |      90 |     138 |     440
%C A105887    Q(N,*)/Q(N) | 0.22045 | 0.26136 | 0.20455 | 0.31364 | 1.00000
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887       N = 10^5 |     753 |     891 |     750 |    1129 |    3523
%C A105887    Q(N,*)/Q(N) | 0.21374 | 0.25291 | 0.21289 | 0.32047 | 1.00000
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887       N = 10^6 |    6153 |    7395 |    6176 |    9247 |   28971
%C A105887    Q(N,*)/Q(N) | 0.21238 | 0.25526 | 0.21318 | 0.31918 | 1.00000
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887       N = 10^7 |   52427 |   62973 |   52368 |   78398 |  246166
%C A105887    Q(N,*)/Q(N) | 0.21297 | 0.25582 | 0.21273 | 0.31848 | 1.00000
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887       N = 10^8 |  453936 |  544551 |  453699 |  680226 | 2132412
%C A105887    Q(N,*)/Q(N) | 0.21287 | 0.25537 | 0.21276 | 0.31899 | 1.00000
%C A105887   -------------+---------+---------+---------+---------+---------
%C A105887    Conjectured | 0.21277 | 0.25532 | 0.21277 | 0.31915 | 1.00000
%C A105887 (End)
%H A105887 Vincenzo Librandi, <a href="/A105887/b105887.txt">Table of n, a(n) for n = 1..1000</a>
%H A105887 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>
%H A105887 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>
%H A105887 <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>
%t A105887 pr=-15; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
%Y A105887 Cf. A005596 (Artin's constant).
%K A105887 nonn
%O A105887 1,1
%A A105887 _N. J. A. Sloane_, Apr 24 2005