A105887 - OEIS

%I #39 Mar 31 2024 15:04:51

%S 2,11,13,29,37,41,43,59,71,73,89,97,101,103,127,131,149,157,163,179,

%T 191,193,239,251,269,281,307,313,337,359,373,389,401,419,431,433,449,

%U 457,461,479,487,491,509,521,523,547,569,577,599,607,613,641,701,719,727,733,757

%N Primes for which -15 is a primitive root.

%C From _Jianing Song_, Jan 27 2019: (Start)

%C All terms except the first are congruent to 7, 11, 13 or 14 modulo 15. If we define

%C   Pi(N,b) = # {p prime, p <= N, p == b (mod 15)};

%C      Q(N) = # {p prime, 2 < p <= N, p in this sequence},

%C 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 Conjecture: if we further define

%C    Q(N,b) = # {p prime, p <= N, p == b (mod 15), p in this sequence},

%C then we have:

%C    Q(N,7) ~ (10/47)*Q(N) ~ ( 80/95)*C*Pi(N,7);

%C   Q(N,11) ~ (12/47)*Q(N) ~ ( 96/95)*C*Pi(N,11);

%C   Q(N,13) ~ (10/47)*Q(N) ~ ( 80/95)*C*Pi(N,13);

%C   Q(N,14) ~ (15/47)*Q(N) ~ (120/95)*C*Pi(N,14).

%C Numeric verification up tp N = 10^8:

%C                |  Q(N,7) | Q(N,11) | Q(N,13) | Q(N,14) |    Q(N)

%C   -------------+---------+---------+---------+---------+---------

%C       N = 10^3 |      14 |      18 |      13 |      19 |      64

%C    Q(N,*)/Q(N) | 0.21875 | 0.28125 | 0.20313 | 0.29688 | 1.00000

%C   -------------+---------+---------+---------+---------+---------

%C       N = 10^4 |      97 |     115 |      90 |     138 |     440

%C    Q(N,*)/Q(N) | 0.22045 | 0.26136 | 0.20455 | 0.31364 | 1.00000

%C   -------------+---------+---------+---------+---------+---------

%C       N = 10^5 |     753 |     891 |     750 |    1129 |    3523

%C    Q(N,*)/Q(N) | 0.21374 | 0.25291 | 0.21289 | 0.32047 | 1.00000

%C   -------------+---------+---------+---------+---------+---------

%C       N = 10^6 |    6153 |    7395 |    6176 |    9247 |   28971

%C    Q(N,*)/Q(N) | 0.21238 | 0.25526 | 0.21318 | 0.31918 | 1.00000

%C   -------------+---------+---------+---------+---------+---------

%C       N = 10^7 |   52427 |   62973 |   52368 |   78398 |  246166

%C    Q(N,*)/Q(N) | 0.21297 | 0.25582 | 0.21273 | 0.31848 | 1.00000

%C   -------------+---------+---------+---------+---------+---------

%C       N = 10^8 |  453936 |  544551 |  453699 |  680226 | 2132412

%C    Q(N,*)/Q(N) | 0.21287 | 0.25537 | 0.21276 | 0.31899 | 1.00000

%C   -------------+---------+---------+---------+---------+---------

%C    Conjectured | 0.21277 | 0.25532 | 0.21277 | 0.31915 | 1.00000

%C (End)

%H Vincenzo Librandi, <a href="/A105887/b105887.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>

%H <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>

%t pr=-15; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

%Y Cf. A005596 (Artin's constant).

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Apr 24 2005