A019334 -id:A019334 - cp's OEIS frontend

A062117 Order of 3 mod n-th prime.

1, 0, 4, 6, 5, 3, 16, 18, 11, 28, 30, 18, 8, 42, 23, 52, 29, 10, 22, 35, 12, 78, 41, 88, 48, 100, 34, 53, 27, 112, 126, 65, 136, 138, 148, 50, 78, 162, 83, 172, 89, 45, 95, 16, 196, 198, 210, 222, 113, 57, 232, 119, 120, 125, 256, 131, 268, 30, 69, 280, 282, 292, 34

Offset: 1

Olivier Gérard, Jun 06 2001

			The 3rd prime is 5 and mod 5, 3^4 = 1, so a(3) = 4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A019334 (full reptend primes in base 3).

In other bases: A014664, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(3,A000040[n])); # Muniru A Asiru, Feb 07 2019

Table[With[{p=Prime[n]},If[p==3,0,MultiplicativeOrder[3,p]]],{n,63}] (* Ray Chandler, Apr 06 2016 *)

a(n,{base=3}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

from sympy import n_order, prime
def A062117(n): return n_order(3,prime(n)) if n != 2 else 0 # Chai Wah Wu, Nov 10 2023

A167792 Numbers with primitive root 3.

Original entry on oeis.org

2, 4, 5, 7, 10, 14, 17, 19, 25, 29, 31, 34, 38, 43, 49, 50, 53, 58, 62, 79, 86, 89, 98, 101, 106, 113, 125, 127, 137, 139, 149, 158, 163, 173, 178, 197, 199, 202, 211, 223, 226, 233, 250, 254, 257, 269, 274, 278, 281, 283, 289, 293, 298, 317, 326, 331, 343, 346

Offset: 1

T. D. Noe, Nov 12 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A019334 (primes with primitive root 3)

pr=3; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

is(n)=if(n%3==0, return(0)); my(p=eulerphi(n)); znorder(Mod(3, n), p)==p \\ Charles R Greathouse IV, Jan 04 2025

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

A001123 Primes with 3 as smallest primitive root.

Original entry on oeis.org

7, 17, 31, 43, 79, 89, 113, 127, 137, 199, 223, 233, 257, 281, 283, 331, 353, 401, 449, 463, 487, 521, 569, 571, 593, 607, 617, 631, 641, 691, 739, 751, 809, 811, 823, 857, 881, 929, 953, 977, 1013, 1039, 1049, 1063, 1087, 1097, 1193, 1217

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, A001124, etc.

Cf. A019334.

Prime[ Select[ Range[200], PrimitiveRoot[ Prime[ # ]] == 3 & ]]
(* or *)
Select[ Prime@Range@200, PrimitiveRoot@# == 3 &] (* Robert G. Wilson v, May 11 2001 *)

forprime(p=3, 1000, if(znorder(Mod(2, p))!=p-1&&znorder(Mod(3, p))==p-1, print1(p,", ")));

{ n=0; forprime (p=3, 99999, if (znorder(Mod(2,p))!=p-1 && znorder(Mod(3,p))==p-1, n++; write("b001123.txt", n, " ", p); if (n>=1000, break) ) ) } \\ Harry J. Smith, Jun 14 2009

from itertools import islice
from sympy import nextprime, is_primitive_root
def A001123_gen(): # generator of terms
    p = 3
    while (p:=nextprime(p)):
        if not is_primitive_root(2,p) and is_primitive_root(3,p):
            yield p
A001123_list = list(islice(A001123_gen(),30)) # Chai Wah Wu, Feb 13 2023

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

A019421 Primes with primitive root 99.

Original entry on oeis.org

2, 17, 23, 41, 47, 59, 67, 71, 73, 101, 103, 109, 149, 173, 179, 191, 197, 199, 223, 233, 241, 251, 277, 293, 311, 331, 337, 349, 367, 373, 379, 383, 409, 419, 443, 457, 461, 467, 499, 541, 557, 569, 587, 593, 599, 613, 619, 631, 643, 673, 677, 683, 691, 719, 733, 761

Offset: 1

David W. Wilson

nonn

To allow primes less than the specified primitive root m (here, 99) 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=99; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

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 &]

A105874 Primes for which -2 is a primitive root.

Original entry on oeis.org

5, 7, 13, 23, 29, 37, 47, 53, 61, 71, 79, 101, 103, 149, 167, 173, 181, 191, 197, 199, 239, 263, 269, 271, 293, 311, 317, 349, 359, 367, 373, 383, 389, 421, 461, 463, 479, 487, 503, 509, 541, 557, 599, 607, 613, 647, 653, 661, 677, 701, 709, 719, 743, 751, 757, 773, 797

Offset: 1

N. J. A. Sloane, Apr 24 2005

nonn

Also primes for which (p-1)/2 (==-1/2 mod p) is a primitive root. [Joerg Arndt, Jun 27 2011]

Joerg Arndt, Table of n, a(n) for n = 1..10000
L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
Index entries for primes by primitive root

Cf. A001122, A019334-A019338, A001913, A019339-A019367 etc., A105875-A105914.

with(numtheory); f:=proc(n) local t1,i,p; t1:=[]; for i from 1 to 500 do p:=ithprime(i); if order(n,p) = p-1 then t1:=[op(t1),p]; fi; od; t1; end; f(-2);

pr=-2; 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}];
Select[Range[400], Reduce[a[#, 1] == 1, Integers] &];
2 % + 1 (* Gerry Martens, Apr 28 2015 *)

forprime(p=3,10^4,if(p-1==znorder(Mod(-2,p)),print1(p", "))); /* Joerg Arndt, Jun 27 2011 */

from sympy import n_order, nextprime
from itertools import islice
def A105874_gen(startvalue=3): # generator of terms >= startvalue
    p = max(startvalue-1,2)
    while (p:=nextprime(p)):
        if n_order(-2,p) == p-1:
            yield p
A105874_list = list(islice(A105874_gen(),20)) # Chai Wah Wu, Aug 11 2023

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 belonging to this sequence when a(p,1)==1. - Gerry Martens, May 21 2015

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

A167408 Orderly numbers: a number n is orderly if there exists some number k > tau(n) such that the set of the divisors of n is congruent to the set {1,2,...,tau(n)} mod k.

Original entry on oeis.org

1, 2, 5, 7, 8, 9, 11, 12, 13, 17, 19, 20, 23, 27, 29, 31, 37, 38, 41, 43, 47, 52, 53, 57, 58, 59, 61, 67, 68, 71, 72, 73, 76, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 118, 124, 127, 131, 133, 137, 139, 149, 151, 157, 158, 162, 163, 164, 167, 173, 177, 178, 179

Offset: 1

Andrew Weimholt, Nov 03 2009

   n: {divisors(n)} == {1,2,...,tau(n)} mod k
  -------------------------------------------
{1} == {1} mod 2
{1,2} == {1,2} mod 3
{1,5} == {1,2} mod 3
{1,7} == {1,2} mod 5
{1,2,8,4} == {1,2,3,4} mod 5
{1,9,3} == {1,2,3} mod 7
{1,11} == {1,2} mod 3 or 9
{1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
{1,13} == {1,2} mod 11
{1,17} == {1,2} mod 3,5, or 15
{1,19} == 1,2 mod 17
{1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
{1,23} == {1,2} mod 3,7, or 21
{1,27,3,9} == {1,2,3,4} mod 5
{1,29} == {1,2} mod 3,9, or 27
{1,31} == {1,2} mod 29
{1,37} == 1,2 mod 5,7, or 35
{1,2,38,19} == {1,2,3,4} mod 5
{1,41} == {1,2} mod 3,13, or 39
{1,43} == {1,2} mod 41
{1,47} == {1,2} mod 3,5,9,15, or 45
{1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
{1,53} == {1,2} mod 3,17, or 51
{1,57,3,19} == {1,2,3,4} mod 5
{1,2,58,29} == {1,2,3,4} mod 5
{1,59} == {1,2} mod 3,19, or 57
{1,61} == {1,2} mod 59
{1,67} == {1,2} mod 5,13, or 65
{1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
{1,71} == {1,2} mod 3,23, or 69
{1,2,3,4,18,6,72,8,9,36,24,12} == {1,2,3,4,5,6,7,8,9,10,11,12} mod 13
{1,73} == {1,2} mod 71
{1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
{1,79} == {1,2} mod 7,11, or 77
{1,83} == {1,2} mod 3,9,27, or 81
{1,87,3,29} == {1,2,3,4} mod 5
{1,89} == {1,2} mod 3,29, or 87
{1,97} == {1,2} mod 5,19, or 95
The primes other than 3 are orderly.
Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7.
For primes, k values can be p-2 or a divisor of p-2 other than 1.
T. D. Noe observed that for composite orderly numbers, n, k seems to be one of the three values: tau(n)+1, tau(n)+3, tau(n)+4.
The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7.
The orderly numbers with k = tau(n)+3 come in many forms. See A168003. It appears that tau(n)+3 is a prime with primitive root 2 (A001122).
The forms for composite orderly numbers with k = tau(n)+1 are too numerous to list here, but seem to occur for any prime k > 3.
Let p be any prime. Then p^(m-2) is in this sequence if m is a prime with primitive root p. For example, 2^(m-2) is here for every m in A001122; 3^(m-2) is here for every m in A019334; 5^(m-2) is here for every m in A019335. For every prime p, there appear to be an infinite number of prime powers p^(m-2) here. All these numbers are actually very orderly (A167409) because we can choose k = tau(n)+1. - T. D. Noe, Nov 04 2009

			12 is an orderly number because 12's divisors are 1,2,3,4,6,12 and
   1 == 1 (mod 7)
   2 == 2 (mod 7)
   3 == 3 (mod 7)
   4 == 4 (mod 7)
  12 == 5 (mod 7)
   6 == 6 (mod 7)

A. Weimholt, Table of n, a(n) for n = 1..10000
Bill McEachen, A167408/A002858

Cf. A167409 = very orderly numbers (k = tau(n) + 1).
Cf. A167410 = disorderly numbers = numbers not in this sequence.
Cf. A167411 = minimal k values for the orderly numbers.

orderlyQ[n_] := (For[dd = Divisors[n]; tau = Length[dd]; k = 3, k <= Max[tau + 4, Last[dd] - 2], k++, If[ Union[ Mod[dd, k]] == Range[tau], Return[True]]]; False); Select[ Range[180], orderlyQ] (* Jean-François Alcover, Aug 19 2013 *)

Minor editing by N. J. A. Sloane, Nov 06 2009

Information about the tau(n)+3 orderly numbers corrected by T. D. Noe, Nov 16 2009

cp's OEIS Frontend

A062117 Order of 3 mod n-th prime.

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A167792 Numbers with primitive root 3.

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A019335 Primes with primitive root 5.

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A001123 Primes with 3 as smallest primitive root.

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A019421 Primes with primitive root 99.

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A019337 Primes with primitive root 7.

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A019339 Primes with primitive root 11.

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A105874 Primes for which -2 is a primitive root.

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Maple