A105887 Primes for which -15 is a primitive root.

2, 11, 13, 29, 37, 41, 43, 59, 71, 73, 89, 97, 101, 103, 127, 131, 149, 157, 163, 179, 191, 193, 239, 251, 269, 281, 307, 313, 337, 359, 373, 389, 401, 419, 431, 433, 449, 457, 461, 479, 487, 491, 509, 521, 523, 547, 569, 577, 599, 607, 613, 641, 701, 719, 727, 733, 757

Offset: 1

Author

N. J. A. Sloane, Apr 24 2005

Keywords

• nonn

Comments

From Jianing Song, Jan 27 2019: (Start)

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

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

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

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.

Conjecture: if we further define

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

then we have:

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

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

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

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

Numeric verification up tp N = 10^8:

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

-------------+---------+---------+---------+---------+---------

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

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

-------------+---------+---------+---------+---------+---------

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

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

-------------+---------+---------+---------+---------+---------

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

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

-------------+---------+---------+---------+---------+---------

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

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

-------------+---------+---------+---------+---------+---------

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

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

-------------+---------+---------+---------+---------+---------

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

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

-------------+---------+---------+---------+---------+---------

Conjectured | 0.21277 | 0.25532 | 0.21277 | 0.31915 | 1.00000

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Artin's constant
• Wikipedia, Artin's conjecture on primitive roots
• Index entries for primes by primitive root

Crossrefs

Cf. A005596 (Artin's constant).

Programs

Mathematica

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