A019335 -id:A019335 - cp's OEIS frontend

A211241 Order of 5 mod n-th prime: least k such that prime(n) divides 5^k-1.

1, 2, 0, 6, 5, 4, 16, 9, 22, 14, 3, 36, 20, 42, 46, 52, 29, 30, 22, 5, 72, 39, 82, 44, 96, 25, 102, 106, 27, 112, 42, 65, 136, 69, 37, 75, 156, 54, 166, 172, 89, 15, 19, 192, 196, 33, 35, 222, 226, 114, 232, 119, 40, 25, 256, 262, 67, 27, 276, 140, 282, 292

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
Alexandre Zalesski, Unisingular subgroups of symplectic group Sp_2n(2) for 2n < 250, arXiv:2401.16075 [math.GR], 2024. See p. 52.

Cf. A019335 (full reptend primes in base 5).

In other bases: A014664, A062117, A082654, A211242, A211243, A211244, A211245, A002371.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(5,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 5; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=5}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

A019334 Primes with primitive root 3.

Original entry on oeis.org

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

A167793 Numbers with primitive root 5.

Original entry on oeis.org

2, 3, 6, 7, 9, 14, 17, 18, 23, 27, 34, 37, 43, 46, 47, 49, 53, 54, 73, 74, 81, 83, 86, 94, 97, 98, 103, 106, 107, 113, 137, 146, 157, 162, 166, 167, 173, 193, 194, 197, 206, 214, 223, 226, 227, 233, 243, 257, 263, 274, 277, 283, 289, 293, 307, 314, 317, 334, 343, 346

Offset: 1

T. D. Noe, Nov 12 2009

nonn

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

Cf. A019335 (primes with primitive root 5)

pr=5; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

is(n)=if(n%5==0, return(0)); my(p=eulerphi(n)); znorder(Mod(5, n), p)==p \\ Charles R Greathouse IV, Jan 04 2025

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

A384948 Primes p == 3 (mod 4) such that 5 is a primitive root of integers modulo p, but 2+-i are not primitive roots of Gaussian integers modulo p.

Original entry on oeis.org

83, 307, 347, 503, 587, 863, 947, 1103, 1223, 1523, 1567, 1667, 1787, 1907, 2063, 2087, 2267, 2663, 2683, 2687, 2903, 2963, 3167, 3343, 3347, 3623, 3803, 3863, 4283, 4463, 4523, 4643, 4967, 5147, 5303, 5387, 5507, 5563, 5807, 5843, 6047, 6203, 6607, 6863, 6983, 7187, 7247, 7523, 7583

Offset: 1

Jianing Song, Jun 20 2025

For p = A002145(k), A385165(k) divides (p+1) * ord(5,p), since we have (2+-i)^(p+1) == 5 (mod p). Hence if 2+-i are primitive roots of Gaussian integers modulo p, then 5 is a primitive root of integers modulo p. This sequence lists p such that the converse does not hold.

			5 is a primitive root modulo 83, but the multiplicative order of 2+-i modulo 83 in Gaussian integers is not 83^2 - 1 = 6888; it is 2296 = 6888/3. In other words, 2+-i are not generators of (Z[i]/83Z[i])*.

Jianing Song, Table of n, a(n) for n = 1..10000

By definition, subsequence of A019335, A122870, and A385168.

isprim(p) = my(f = factor(p^2-1)[,1]~); for(i=1, #f, if(Mod([2, -1; 1, 2], p)^((p^2-1)/f[i]) == 1, return(0))); return(1) \\ for a prime p == 3 (mod 4), determines if 2+-i are primitive roots modulo p
isA384948(p) = isprime(p) && (p%4==3) && znorder(Mod(5,p))==p-1 && !isprim(p)

A071522 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(5).

Original entry on oeis.org

1, 2, 6, 16, 22, 36, 42, 46, 52, 72, 82, 96, 102, 106, 112, 136, 156, 166, 172, 192, 196, 222, 226, 232, 256, 262, 276, 282, 292, 306, 316, 346, 352, 372, 382, 396, 432, 442, 462, 466, 502, 522, 546, 556, 562, 576, 586, 592, 606, 612, 616, 646, 652, 672, 676

Offset: 1

Robert G. Wilson v, Jun 22 2002

Numbers k = p - 1 such that p is a prime with primitive root 5. - Joerg Arndt, Jun 25 2020

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

Cf. A071642. Contained in A006093.

filter:= proc(n)
isprime(n+1) and numtheory:-order(5,n+1)=n
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 25 2020

Select[Prime[Range[1000]], MultiplicativeOrder[5, #] == # - 1&] - 1 (* Jean-François Alcover, Aug 16 2020 *)

forprime(p=2, 10^3, if(p==5,next()); if(znorder(Mod(5,p))==p-1, print1(p-1,", "))); \\ Joerg Arndt, Jun 25 2020

a(n) = A019335(n) - 1. - Joerg Arndt, Jun 25 2020

a(1)=1 inserted by Robert Israel, Jun 24 2020

A103491 Multiplicative suborder of 5 (mod n) = sord(5, n).

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 3, 2, 3, 0, 5, 2, 2, 3, 0, 4, 8, 3, 9, 0, 3, 5, 11, 2, 0, 2, 9, 6, 7, 0, 3, 8, 10, 8, 0, 6, 18, 9, 4, 0, 10, 3, 21, 5, 0, 11, 23, 4, 21, 0, 16, 4, 26, 9, 0, 6, 18, 7, 29, 0, 15, 3, 3, 16, 0, 10, 11, 16, 11, 0, 5, 6, 36, 18, 0, 9, 30, 4, 39, 0, 27, 10, 41, 6, 0, 21, 7, 10, 22, 0

Offset: 0

Harry J. Smith, Feb 08 2005

a(n) is minimum e for which 5^e = +/-1 mod n, or zero if no e exists.

For n > 2, a(n) <= (n-1)/2, with equality if (but not only if) n is in A019335. - Robert Israel, Mar 20 2020

H. Cohen, Course in Computational Algebraic Number Theory, Springer, 1993, p. 25, Algorithm 1.4.3

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Multiplicative Order.
S. Wolfram, Algebraic Properties of Cellular Automata (1984), Appendix B.
Eric Weisstein's World of Mathematics, Suborder Function

Cf. A019335.

f:= proc(n) local x;
  if n mod 5 = 0 then return 0 fi;
  x:= numtheory:-mlog(-1,5,n);
  if x <> FAIL then x else numtheory:-order(5,n) fi
end proc:
f(1):= 0:
map(f, [$0..100]); # Robert Israel, Mar 20 2020

Suborder[k_, n_] := If[n > 1 && GCD[k, n] == 1, Min[MultiplicativeOrder[k, n, {-1, 1}]], 0];
a[n_] := Suborder[5, n];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 21 2020, after T. D. Noe in A003558 *)

A241044 Primes having primitive roots 2, 3, and 5.

Original entry on oeis.org

53, 173, 197, 293, 317, 557, 653, 677, 773, 797, 907, 1277, 1373, 1483, 1493, 1637, 1733, 1747, 1987, 1997, 2083, 2213, 2237, 2333, 2357, 2467, 2477, 2683, 2693, 2837, 2957, 3307, 3413, 3533, 3547, 3557, 3643, 3677, 3797, 3917, 4003, 4013, 4133, 4157

Offset: 1

T. D. Noe, Apr 16 2014

nonn

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

Cf. A001122, A019334, A019335, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[600]], fQ[2, #] && fQ[3, #] && fQ[5, #] &]
Select[Prime[Range[600]],SequenceCount[PrimitiveRootList[#],{2,3,5}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 03 2018 *)

A241045 Primes having primitive roots 2, 3, 5, and 7.

Original entry on oeis.org

173, 293, 677, 773, 797, 907, 1277, 1637, 1747, 2083, 2357, 2477, 2693, 2957, 3533, 3797, 4133, 4157, 4373, 4493, 4603, 4637, 4877, 4973, 5333, 5477, 5717, 5813, 5923, 6053, 6173, 6317, 6547, 6653, 6763, 7013, 7517, 8237, 8573, 8693, 8837, 9173, 9533

Offset: 1

T. D. Noe, Apr 16 2014

nonn

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

Cf. A001122, A019334, A019335, A019337, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[1200]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] &]
Select[Prime[Range[1200]],SubsetQ[PrimitiveRootList[#],{2,3,5,7}]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 16 2020 *)

A241047 Primes having primitive roots 2, 3, 5, 7, 11, and 13.

Original entry on oeis.org

293, 2477, 4373, 6173, 7013, 9173, 9677, 10853, 13037, 13397, 13613, 13877, 14957, 15413, 17093, 17597, 18413, 18917, 19157, 22277, 22613, 24317, 26813, 27653, 27893, 29333, 30197, 31517, 33893, 34613, 34877, 35573, 37253, 40493, 41117, 41333, 42437

Offset: 1

T. D. Noe, Apr 16 2014

nonn

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

Cf. A001122, A019334, A019335, A019337, A019339, A019341, A213052.

fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[4500]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] &]

cp's OEIS Frontend

A211241 Order of 5 mod n-th prime: least k such that prime(n) divides 5^k-1.

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A019334 Primes with primitive root 3.

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A167793 Numbers with primitive root 5.

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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.

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A384948 Primes p == 3 (mod 4) such that 5 is a primitive root of integers modulo p, but 2+-i are not primitive roots of Gaussian integers modulo p.

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A071522 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(5).

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A103491 Multiplicative suborder of 5 (mod n) = sord(5, n).

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A241044 Primes having primitive roots 2, 3, and 5.

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