A001123 -id:A001123 - cp's OEIS frontend

A001122 Primes with primitive root 2.

3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797

Offset: 1

N. J. A. Sloane

Artin conjectured that this sequence is infinite.
Conjecture: sequence contains infinitely many pairs of twin primes. - Benoit Cloitre, May 08 2003
Pieter Moree writes (Oct 20 2004): Assuming the Generalized Riemann Hypothesis, it can be shown that the density of primes p such that a prescribed integer g has order (p-1)/t, with t fixed, exists and, moreover, it can be computed. This density will be a rational number times the so-called Artin constant. For 2 and 10 the density of primitive roots is A, the Artin constant itself.
It seems that this sequence consists of A050229 \ {1,2}.
Primes p such that 1/p, when written in base 2, has period p-1, which is the greatest period possible for any integer.
Positive integer 2*m-1 is in the sequence iff A179382(m)=m-1. - Vladimir Shevelev, Jul 14 2010
These are the odd primes p for which the polynomial 1+x+x^2+...+x^(p-1) is irreducible over GF(2). - V. Raman, Sep 17 2012 [Corrected by N. J. A. Sloane, Oct 17 2012]
Prime(n) is in the sequence if (and conjecturally only if) A133954(n) = prime(n). - Vladimir Shevelev, Aug 30 2013
Pollack shows that, on the GRH, that there is some C such that a(n+1) - a(n) < C infinitely often (in fact, 1 can be replaced by any positive integer). Further, for any m, a(n), a(n+1), ..., a(n+m) are consecutive primes infinitely often. - Charles R Greathouse IV, Jan 05 2015
From Jianing Song, Apr 27 2019: (Start)
All terms are congruent to 3 or 5 modulo 8. If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 8)};
     Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,3) + Pi(N,5)), where C = A005596 is Artin's constant.
Conjecture: if we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 8), p in this sequence},
then we have:
   Q(N,3) ~ (1/2)*Q(N) ~ C*Pi(N,3);
   Q(N,5) ~ (1/2)*Q(N) ~ C*Pi(N,5). (End)
Conjecture: for a prime p > 5, p has primitive root 2 iff p == +-3 (mod 8) divides 2^k + 3 for some k < p - 1 and divides 2^m + 5 for some m < p - 1. It seems that all primes of the form 2^k + 3 for k <> 2 (A057732) have primitive root 2. - Thomas Ordowski, Nov 27 2023

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I; see p. 221.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 169.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 56.
Lehmer, D. H. and Lehmer, Emma; Heuristics, anyone? in Studies in mathematical analysis and related topics, pp. 202-210, Stanford Univ. Press, Stanford, Calif., 1962.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 20.
D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Joerg Arndt, Matters Computational (The Fxtbook), pp. 876-878.
Richard Bartels, Generalized Loewy Length of Cohen-Macaulay Local and Graded Rings, arXiv:2308.14932 [math.AC], 2023. See p. 10.
J. Conde, M. Miller, J. M. Miret, and K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, 9(2) (2015), 145-149.
Jonathan Detchart and Jérôme Lacan, Improving the coding speed of erasure codes with polynomial ring transforms, arXiv:1709.00178 [cs.IT], 2017.
K. Dilcher and L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, 22 (2014), 77-102.
Gabriele Fici and Estéban Gabory, Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform, arXiv:2502.12844 [math.CO], 2025. See p. 12.
R. Gupta and M. R. Murty, A remark on Artin's conjecture, Invent. Math. 78 (1984), 127-230.
C. Hooley, On Artin's conjecture, J. Reine Angewandte Math., 225 (1967), 209-220.
Florian Ingels, Anaïs Denis, and Bastien Cazaux, Decomposing Words for Enhanced Compression: Exploring the Number of Runs in the Extended Burrows-Wheeler Transform, arXiv:2506.04926 [cs.DS], 2025. See p. 15.
Robert Jackson, Dmitriy Rumynin and Oleg V. Zaboronski, An approach to RAID-6 based on cyclic groups, Applied Mathematics & Information Sciences, 5(2) (2011), 148-170.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.
Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents, IACR, Report 2012/033, 2012.
F. Pillichshammer, Bounds for the quality parameter of digital shift nets over Z_2, Finite Fields Applic., 8 (2002), 444-454.
Pieter Moree, Artin's primitive root conjecture-a survey, arXiv:math/0412262 [math.NT], 2004-2012.
Paul Pollack, Bounded gaps between primes with a given primitive root, arXiv:1404.4007 [math.NT], 2014.
Vladimir Shevelev, On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2, arXiv:0710.1354 [math.NT], 2007.
Stephan Tornier, Groups Acting on Trees With Prescribed Local Action, arXiv:2002.09876 [math.GR], 2020.
Qifu Tyler Sun, Hanqi Tang, Zongpeng Li, Xiaolong Yang, and Keping Long, Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths, arXiv:1806.04635 [cs.IT], 2018.
Eric Weisstein's World of Mathematics, Artin's constant.
Wikipedia, Artin's conjecture on primitive roots.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001123, A001913, A001917, A005596 (Artin's constant), A050229, A071642, A127209, A163782, A292270.
Cf. A002326 for the multiplicative order of 2 mod 2n+1. (Alternatively, the least positive value of m such that 2n+1 divides 2^m-1).
Cf. A216838 (Odd primes for which 2 is not a primitive root).

Select[ Prime@Range@200, PrimitiveRoot@# == 2 &] (* Robert G. Wilson v, May 11 2001 *)
pr = 2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == # - 1 &] (* N. J. A. Sloane, Jun 01 2010 *)

forprime(p=3, 1000, if(znorder(Mod(2, p))==(p-1), print1(p,", "))); \\ [corrected by Michel Marcus, Oct 08 2014]

from itertools import islice
from sympy import nextprime, is_primitive_root
def A001122_gen(): # generator of terms
    p = 2
    while (p:=nextprime(p)):
        if is_primitive_root(2,p):
            yield p
A001122_list = list(islice(A001122_gen(),30)) # Chai Wah Wu, Feb 13 2023

Delta(a(n),2^a(n)*x) = a(n)*Delta(a(n),2*x), where Delta(k,x) is the difference between numbers of evil(A001969) and odious(A000069) integers divisible by k in interval [0,x). - Vladimir Shevelev, Aug 30 2013

For n >= 2, a(n) = 1 + 2*A163782(n-1). - Antti Karttunen, Oct 07 2017

A001124 Primes with 5 as smallest primitive root.

Original entry on oeis.org

23, 47, 73, 97, 103, 157, 167, 193, 263, 277, 307, 383, 397, 433, 503, 577, 647, 673, 683, 727, 743, 863, 887, 937, 967, 983, 1033, 1093, 1103, 1153, 1163, 1223, 1367, 1487, 1543, 1583, 1607, 1777, 1823, 1847, 1933, 1993, 2003, 2017, 2063, 2087, 2113, 2203, 2207

Offset: 1

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 57.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for primes by primitive root

Cf. A001122, A001123, A001125, etc.

<< NumberTheory`NumberTheoryFunctions`; Prime[ Select[ Range[200], PrimitiveRoot[ Prime[ # ] ] == 5 & ] ]
(* first load *) << NumberTheory`NumberTheoryFunctions` (* then *) Select[ Prime@Range@300, PrimitiveRoot@# == 5 &] (* Robert G. Wilson v, May 11 2001 *)
Select[Prime[Range[350]],PrimitiveRoot[#]==5&] (* The PrimitiveRoot function is now part of Mathematica's core, so no add-in needs to be loaded before calling it *) (* Harvey P. Dale, Dec 06 2014 *)

from itertools import islice
from sympy import nextprime, primitive_root
def A001124_gen(): # generator of terms
    p = 5
    while (p:=nextprime(p)):
        if primitive_root(p)==5:
            yield p
A001124_list = list(islice(A001124_gen(),30)) # Chai Wah Wu, Feb 13 2023

More terms from Robert G. Wilson v, May 10 2001

A066529 a(n) is the least index such that the least primitive root of the a(n)-th prime is n, or zero if no such prime exists.

Original entry on oeis.org

1, 2, 4, 0, 9, 13, 20, 0, 0, 65, 117, 566, 88, 173, 85, 0, 64, 5426, 43, 10217, 80, 474, 326, 44110, 0, 1479, 0, 12443, 1842, 11662, 775, 0, 23559, 5029, 6461, 0, 3894, 5629, 15177, 105242, 14401, 182683, 9204, 7103, 5518399, 23888, 24092, 42304997, 0, 1455704, 27848, 12107, 14837, 205691645, 38451, 12102037, 39370, 28902, 57481, 56379, 90901, 53468, 5918705, 0, 732055, 1738826, 242495, 265666, 10523, 388487, 260680

Offset: 1

Wouter Meeussen, Jan 06 2002

nonn

The corresponding primes are in A023048.

For n < 150, only a(108) is presently unknown. - Robert G. Wilson v, Jan 03 2006

			a(6) = 13 because Prime[13] = 41 is the least prime with least primitive root = 6

Tomás Oliveira e Silva, Least prime primitive root of prime numbers
E. Weisstein, Primitive Roots
Index entries for primes by primitive root

Cf. A001918, A001122, A001123, A023048, A001597.

big = Table[ p = Prime[ n ]; PrimitiveRoot[ p ], {n, 1, 1000000} ]; Flatten[ Table[ Position[ big, n, 1, 1 ]/.{}-> 0, {n, 79} ] ] (* First load package NumberTheory`NumberTheoryFunctions` *)
(* first load package *) << NumberTheory`NumberTheoryFunctions` (* then do *) t = Table[0, {100}]; Do[a = PrimitiveRoot@Prime@n; If[a < 101 && t[[a]] == 0, t[[a]] = n], {n, 10^6}]; t (* Robert G. Wilson v, Dec 15 2005 *)

a(n) = 0 iff n is a perfect power (A001597) > 1. - Robert G. Wilson v, Jan 03 2006

a(n) = min { k | A001918(k) = n } U {0} = A000720(A023048(n)) (or zero). - M. F. Hasler, Jun 01 2018

Edited by Dean Hickerson, Jan 14 2002

Further terms from Robert G. Wilson v, Jan 03 2006

A001125 Primes with 6 as smallest primitive root.

Original entry on oeis.org

41, 109, 151, 229, 251, 271, 367, 733, 761, 971, 991, 1069, 1289, 1303, 1429, 1471, 1759, 1789, 1811, 1879, 2411, 2441, 2551, 2749, 2791, 3061, 3079, 3109, 3229, 3251, 3301, 3319, 3967, 4211, 4549, 4721, 4783, 4909, 4931, 4951, 5101, 5167, 5581, 5791

Offset: 1

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 11.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for primes by primitive root

Cf. A001122, A001123, A001124, etc.

Prime[ Select[ Range[1000], PrimitiveRoot[ Prime[ # ] ] == 6 & ] ]
(* or *)
Select[ Prime@Range@760, PrimitiveRoot@# == 6 &] (* Robert G. Wilson v, May 11 2001 *)

More terms from Robert G. Wilson v, May 10 2001

A072190 Indices of primes with primitive root 2.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 12, 16, 17, 18, 19, 23, 26, 28, 32, 34, 35, 38, 40, 41, 42, 45, 47, 49, 57, 62, 66, 69, 70, 74, 75, 77, 81, 82, 86, 89, 91, 94, 97, 99, 100, 101, 102, 103, 107, 112, 114, 119, 120, 121, 123, 126, 127, 134, 137, 138, 139, 142, 144, 145, 147

Offset: 1

Miklos Kristof, Jul 02 2002

Artin conjectured that this sequence is infinite (this is the famous Artin Conjecture).

			8 is an element of the sequence: 19 the 8th prime and 2 is primitive root of 19. 9 is not element of the sequence, since 23 is the 9th prime and 2 is not primitive root of 23.

M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996. see p. 169
L. Huber, manuscripts on Group Theory and Number Theory, 1990-1995

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Artin's conjecture

Cf. A001122, A001123, A001124, A001125.

Select[Range[300], MultiplicativeOrder[2, Prime[#]] == Prime[#] - 1 &] (* T. D. Noe, Apr 16 2014 *)

Edited by N. J. A. Sloane, Apr 11 2009

Extended by T. D. Noe, Apr 16 2014

A225888 Primes p such that neither 2 nor 3 are primitive roots, but together 2 and 3 generate the nonzero residues mod p.

Original entry on oeis.org

41, 103, 109, 151, 157, 229, 251, 271, 277, 367, 397, 683, 733, 761, 967, 971, 991, 1051, 1069, 1163, 1181, 1289, 1303, 1429, 1471, 1543, 1759, 1783, 1789, 1811, 1879, 2003, 2297, 2411, 2441, 2551, 2749, 2791, 2887, 2917, 3061, 3079, 3109, 3229, 3251, 3301, 3319

Offset: 1

John L. Drost, May 19 2013

nonn

			2 has multiplicative order 20 mod 41, 3 has order 8 mod 41 so neither is a primitive root. The subgroup 2 and 3 generate together will have order lcm(20,8) = 40 so 2 and 3 generate all nonzero residues.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001122, A001123.

is(n)=if(n>40 && isprime(n), my(a=znorder(Mod(2,n)),b); if(a==n-1,return(0)); b=znorder(Mod(3,n)); bCharles R Greathouse IV, May 19 2013

A241041 Differences between primes having smallest primitive root 3.

Original entry on oeis.org

10, 14, 12, 36, 10, 24, 14, 10, 62, 24, 10, 24, 24, 2, 48, 22, 48, 48, 14, 24, 34, 48, 2, 22, 14, 10, 14, 10, 50, 48, 12, 58, 2, 12, 34, 24, 48, 24, 24, 36, 26, 10, 14, 24, 10, 96, 24, 14, 48, 48, 34, 48, 14, 10, 14, 12, 22, 72, 14, 12, 22, 12, 14, 36, 34, 2

Offset: 1

T. D. Noe, Apr 16 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Pollack, Bounded gaps between primes with a given primitive root, arXiv 1404.4007, April 15, 2014.

Cf. A001123 (primes having smallest primitive root 3).

Differences[Select[Prime[Range[1000]], PrimitiveRoot[#] == 3 &]]

A241046 Primes having primitive roots 2, 3, 5, 7, and 11.

Original entry on oeis.org

173, 293, 677, 2083, 2477, 3533, 3797, 4133, 4157, 4373, 4603, 4637, 5477, 5717, 5923, 6173, 7013, 9173, 9533, 9677, 10853, 11587, 12437, 13037, 13397, 13613, 13877, 14717, 14957, 15077, 15413, 16253, 17093, 17573, 17597, 18413, 18773, 18917, 19157, 19997

Offset: 1

T. D. Noe, Apr 16 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001122, A001123, A001124, A001126, A019339, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[2300]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] &]

A280930 Primes for which 3 is not the smallest primitive root.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 83, 97, 101, 103, 107, 109, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 229, 239, 241, 251, 263, 269, 271, 277, 293, 307, 311, 313, 317, 337, 347, 349, 359, 367, 373

Offset: 1

Vincenzo Librandi, Jan 11 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..7700

Complement of A001123.

Select[Prime[Range[1, 100]], PrimitiveRoot[#] =!= 3 &]
Select[Prime[Range[100]],PrimitiveRoot[#,1]!=3&] (* Harvey P. Dale, Jan 01 2018 *)

A219461 Primes p such that 2^p - 1 is a prime with primitive root 3.

Original entry on oeis.org

3, 5, 7, 17, 19, 89, 107, 521, 2203, 2281, 4253

Offset: 1

Arkadiusz Wesolowski, Nov 20 2012

If A096393(n) = 3, then A000043(n) belongs to this sequence.
2^a(n) - 1 is in A001123.
a(12) >= 21701. - Max Alekseyev, Feb 08 2024

Discussion titled Conjecture about multiplicative order of 3 modulo a Mersenne prime, (postings in mersenneforum.org), starting May 14 2006.
Index entries for primes by primitive root

Subsequence of A000043.

Cf. A001123, A096393.

Select[Prime@Range[98], PrimitiveRoot[2^# - 1] == 3 &]

a(9)-a(11) from Max Alekseyev, Feb 07 2024

cp's OEIS Frontend

A001122 Primes with primitive root 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A001124 Primes with 5 as smallest primitive root.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A066529 a(n) is the least index such that the least primitive root of the a(n)-th prime is n, or zero if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A001125 Primes with 6 as smallest primitive root.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Extensions

A072190 Indices of primes with primitive root 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A225888 Primes p such that neither 2 nor 3 are primitive roots, but together 2 and 3 generate the nonzero residues mod p.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A241041 Differences between primes having smallest primitive root 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A241046 Primes having primitive roots 2, 3, 5, 7, and 11.

Views

Author

Keywords