A019421 -id:A019421 - cp's OEIS frontend

A019334 Primes with primitive root 3.

2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, 269, 281, 283, 293, 317, 331, 353, 379, 389, 401, 449, 461, 463, 487, 509, 521, 557, 569, 571, 593, 607, 617, 631, 641, 653, 677, 691, 701, 739, 751, 773, 797

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 3) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019
From Jianing Song, Apr 27 2019: (Start)
All terms except the first are congruent to 5 or 7 modulo 12. If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 12)};
     Q(N) = # {p prime, 2 < p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,5) + Pi(N,7)), where C = A005596 is Artin's constant.
If we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 12), p in this sequence},
then we have:
   Q(N,5) ~ (3/5)*Q(N) ~ (12/5)*C*Pi(N,5);
   Q(N,7) ~ (2/5)*Q(N) ~ ( 8/5)*C*Pi(N,7).
For example, for the first 1000 terms except for a(1) = 2, there are 593 terms == 5 (mod 12) and 406 terms == 7 (mod 12). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, International Conference on Mathematical Computer Engineering (ICMCE-13), pp. 2-7, At VIT University, Chennai, Volume: I, 2013.
J. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, June 2015, Volume 9, Issue 2, pp 145-149.
Eric Weisstein's World of Mathematics, Artin's constant
Wikipedia, Artin's conjecture on primitive roots
Index entries for primes by primitive root

Cf. A005596, A001122 (primitive root 2).

Cf. A019335-A019421.

pr=3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

isok(p) = isprime(p) && (p!=3) && (znorder(Mod(3, p))+1 == p); \\ Michel Marcus, May 12 2019

A019335 Primes with primitive root 5.

Original entry on oeis.org

2, 3, 7, 17, 23, 37, 43, 47, 53, 73, 83, 97, 103, 107, 113, 137, 157, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 317, 347, 353, 373, 383, 397, 433, 443, 463, 467, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 617, 647, 653, 673, 677, 683, 727

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 5) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019
Appears to be the numbers k such that the sequence 5^n mod k has period length k-1. All terms are congruent to 2 or 3 mod 5. - Gary Detlefs, May 21 2014
From Jianing Song, Apr 27 2019: (Start)
If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 5)};
     Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ (20/19)*C*N/log(N) ~ (40/19)*C*(Pi(N,2) + Pi(N,3)), where C = A005596 is Artin's constant.
Conjecture: if we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 5), p in this sequence},
then we have:
   Q(N,2) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,2);
   Q(N,3) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,3). (End)

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

Cf. A019334-A019421.

pr=5; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

isok(p) = isprime(p) && (p != 5) && (znorder(Mod(5, p)) == p-1); \\ Michel Marcus, Apr 27 2019

A019337 Primes with primitive root 7.

Original entry on oeis.org

2, 5, 11, 13, 17, 23, 41, 61, 67, 71, 79, 89, 97, 101, 107, 127, 151, 163, 173, 179, 211, 229, 239, 241, 257, 263, 269, 293, 347, 349, 359, 379, 397, 431, 433, 443, 461, 491, 499, 509, 521, 547, 577, 593, 599, 601, 631, 659, 677, 683, 733, 739, 743, 761, 773, 797, 823

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 7) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 03 2019

All terms apart from the first are == 5, 11, 13, 15, 17, 23 (mod 28) since 7 is a quadratic residue modulo any other prime. By Artin's conjecture, this sequence contains about 37.395% of all primes, that is, about 74.79% of all primes == 5, 11, 13, 15, 17, 23 (mod 28). - Jianing Song, Sep 05 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

Cf. A167795.

pr=7; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

A019339 Primes with primitive root 11.

Original entry on oeis.org

2, 3, 13, 17, 23, 29, 31, 41, 47, 59, 67, 71, 73, 101, 103, 109, 149, 163, 173, 179, 197, 223, 233, 251, 277, 281, 293, 331, 367, 373, 383, 419, 443, 461, 463, 467, 487, 499, 557, 569, 587, 593, 599, 601, 613, 619, 643, 647, 673, 677, 683, 701, 719, 761, 769, 809, 821

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 11) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 03 2019

This is a subsequence of A038882. - Klaus Purath, Jul 03 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

Cf. A071566.

pr=11; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

A019338 Primes with primitive root 8.

Original entry on oeis.org

3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 701, 773, 797, 821, 827, 941, 947, 1019, 1061, 1091, 1109, 1187, 1229, 1259, 1277, 1283, 1301, 1307, 1373, 1427

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 8) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 03 2019
Members of A001122 that are not congruent to 1 mod 3. - Robert Israel, Aug 12 2014
Terms greater than 3 are congruent to 5 or 11 modulo 24. - Jianing Song, May 12 2024 [Corrected on May 13 2025]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

select(t -> isprime(t) and numtheory:-order(8,t) = t-1, [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

pr=8; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *)
a[p_, q_]:=Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))],{n,1,2 q p}]
2 Select[Range[800],Rationalize[N[a[#, 3],20]]==1 &]+1
(* Gerry Martens, Apr 28 2015 *)
Join[{3,5},Select[Prime[Range[250]],PrimitiveRoot[#,8]==8&]] (* Harvey P. Dale, Aug 10 2019 *)

is(n)=isprime(n) && n>2 && znorder(Mod(8,n))==n-1 \\ Charles R Greathouse IV, May 21 2015

Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime of this sequence when a(p,3)==1. - Gerry Martens, May 15 2015

On Artin's conjecture, a(n) ~ (5/3A) n log n, where A = A005596 is Artin's constant. - Charles R Greathouse IV, May 21 2015

A056619 Smallest prime with primitive root n, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 2, 0, 2, 11, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 11, 2, 3, 2, 19, 2, 0, 2, 3, 2, 7, 2, 5, 2, 3, 2, 11, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 19, 2, 3, 2, 0, 2, 7, 2, 3, 2, 19, 2, 5, 2, 3, 2, 13, 2, 5, 2, 3, 2, 5, 2, 11, 2, 3, 2, 5, 2, 11, 2, 3, 2, 7, 2, 7, 2, 3, 2

Offset: 1

Robert G. Wilson v, Aug 07 2000

nonn

a(n) > n/2 for n in { 2, 6, 10, 34 }. Are there any other such indices n? - M. F. Hasler, Feb 21 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Here the primitive root may be larger than the prime, whereas in A023049 it may not be.

Cf. A001122, A001913, A019334-A019421.

f:= proc(n) local p;
   if n::odd then return 2
   elif issqr(n) then return 0
   fi;
   p:= 3;
   do
      if numtheory:-order(n,p) = p-1 then return p fi;
      p:= nextprime(p);
   od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 21 2017

a[n_] := Module[{p}, If[OddQ[n], Return[2], If[IntegerQ[Sqrt[n]], Return[0], p = 3; While[True, If[MultiplicativeOrder[n, p] == p-1, Return[p]]; p = NextPrime[p]]]]];
Array[a, 100] (* Jean-François Alcover, Apr 10 2019, after Robert Israel *)

A056619(n)=forprime(p=2,n*2,gcd(n,p)==1&&znorder(Mod(n,p))==p-1&&return(p)) \\ or, more efficient:
A056619(n)=if(bittest(n,0),2,!issquare(n)&&forprime(p=3,n*2,gcd(n,p)==1&&znorder(Mod(n,p))==p-1&&return(p))) \\ M. F. Hasler, Feb 21 2017

a(n) = 0 only for perfect squares, A000290.

a(n) = 2 for all odd n. a(n) = 0 for even squares. a(n) = 3 for n = 2 (mod 6). a(n) = 5 for n in {12, 18, 22, 28} (mod 30). - M. F. Hasler, Feb 21 2017

Corrected and extended by Jud McCranie, Mar 21 2002

Corrected by Robert Israel, Feb 21 2017

A019336 Primes with primitive root 6.

Original entry on oeis.org

11, 13, 17, 41, 59, 61, 79, 83, 89, 103, 107, 109, 113, 127, 131, 137, 151, 157, 179, 199, 223, 227, 229, 233, 251, 257, 271, 277, 347, 367, 373, 397, 401, 419, 443, 449, 467, 487, 491, 521, 563, 569, 587, 593, 613, 641, 659, 661, 683, 709, 733, 757, 761, 809, 823, 827

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 6) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 03 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

pr=6; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

A019340 Primes with primitive root 12.

Original entry on oeis.org

5, 7, 17, 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 223, 257, 269, 281, 283, 293, 317, 353, 367, 379, 389, 401, 449, 461, 509, 523, 547, 557, 569, 571, 593, 607, 617, 619, 631, 641, 653, 691, 701, 739, 751, 761, 773, 787, 797, 809, 821

Offset: 1

David W. Wilson

To allow primes less than the specified primitive root m (here, 12) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 03 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

pr=12; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

cp's OEIS Frontend

A019334 Primes with primitive root 3.

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A019335 Primes with primitive root 5.

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A019337 Primes with primitive root 7.

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A019339 Primes with primitive root 11.

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A019338 Primes with primitive root 8.

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A056619 Smallest prime with primitive root n, or 0 if no such prime exists.

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A019336 Primes with primitive root 6.

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A019340 Primes with primitive root 12.

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