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

1,1

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

Mathematica

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

Crossrefs

Cf. A005596 (Artin's constant).

Sequence in context: A063587 A038972 A242776 * A019407 A075781 A023288

Adjacent sequences: A105884 A105885 A105886 * A105888 A105889 A105890

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2005

Status

approved