A005596 -id:A005596 - cp's OEIS frontend

A007355 Incorrect duplicate of A297408.

2, 13, 19, 6173, 6299, 6353, 6389, 16057, 16369, 16427, 16883, 17167, 17203, 17257, 18169, 18517, 18899, 20353, 20369, 20593, 20639, 20693, 20809, 22037, 22109, 22153

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

dead

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

A019353 Primes with primitive root 27.

Original entry on oeis.org

2, 5, 17, 29, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977, 1013, 1049, 1061, 1097, 1109, 1193, 1217, 1229, 1277, 1301

Offset: 1

David W. Wilson

nonn

From Jianing Song, May 12 2024: (Start)
Members of A019334 that are not congruent to 1 mod 3. Terms greater than 2 are congruent to 5 modulo 12.
According to Artin's conjecture, the number of terms <= N is roughly ((3/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Artin's conjecture on primitive roots
Index entries for primes by primitive root

Cf. A019334, A005596, A000720, A323594.

pr=27; Select[Prime[Range[300]], MultiplicativeOrder[pr, # ] == #-1 &]

isA019353(n) = isprime(n) && (n!=3) && znorder(Mod(27,n)) == n-1 \\ Jianing Song, May 12 2024

A049229 Primes p such that p-2 is not squarefree.

Original entry on oeis.org

11, 29, 47, 83, 101, 127, 137, 149, 173, 191, 227, 263, 277, 281, 317, 353, 389, 443, 461, 479, 509, 541, 569, 577, 587, 607, 641, 659, 677, 727, 821, 827, 839, 857, 877, 911, 929, 947, 977, 983, 1019, 1031, 1091, 1109, 1129, 1163, 1181, 1217, 1277, 1289

Offset: 1

Labos Elemer

nonn

This sequence is infinite and its relative density in the sequence of the primes is equal to 1 - 2 * Product_{p prime} (1-1/(p*(p-1))) = 1 - 2 * A005596 = 0.252088... - Amiram Eldar, Feb 27 2021

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005596, A013929, A049231, A049232.

Select[Prime[Range[2,300]],!SquareFreeQ[#-2]&] (* Harvey P. Dale, Nov 14 2012 *)

isok(p) = (p>2) && isprime(p) && !issquarefree(p-2); \\ Michel Marcus, May 14 2018

A091880 A049232 indexed by A000040.

Original entry on oeis.org

1, 4, 9, 14, 15, 18, 21, 22, 25, 36, 39, 40, 48, 53, 59, 65, 67, 70, 72, 73, 74, 82, 85, 88, 99, 101, 110, 114, 122, 125, 127, 129, 130, 137, 143, 146, 147, 155, 158, 168, 174, 177, 180, 181, 188, 194, 198, 200, 202, 204, 213, 216, 219, 220, 224, 226, 229, 235

Offset: 1

Ray Chandler, Feb 15 2004

The asymptotic density of this sequence is 1 - 2 * Product_{p prime} (1-1/(p*(p-1))) = 1 - 2 * A005596 = 0.252088... - Amiram Eldar, Feb 28 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A005596, A049232.

Select[Range[100], ! SquareFreeQ[Prime[ # ] + 2] &] (* Zak Seidov, Oct 28 2008 *)

a(n) = k such that A000040(k) = A049232(n).

A103362 Reduced numerators of the fraction of primes < 10^n that are full reptend primes.

Original entry on oeis.org

1, 9, 5, 467, 3617, 14750, 248881, 2155288, 19016617, 170169241

Offset: 1

Eric W. Weisstein, Feb 02 2005

A103362/A103363 is conjectured to approach Artin's constant A005596.

			1/4, 9/25, 5/14, 467/1229, 3617/9592, 14750/39249, ...

Eric Weisstein's World of Mathematics, Full Reptend Prime

Cf. A005596, A006880, A086018, A103363.

Numerator(A086018/A006880).

A103363 Reduced denominators of the fraction of primes < 10^n that are full reptend primes.

Original entry on oeis.org

4, 25, 14, 1229, 9592, 39249, 664579, 5761455, 50847534, 455052511

Offset: 1

Eric W. Weisstein, Feb 02 2005

A103362/A103363 is conjectured to approach Artin's constant A005596.

			1/4, 9/25, 5/14, 467/1229, 3617/9592, 14750/39249, ...

Eric Weisstein's World of Mathematics, Full Reptend Prime

Cf. A005596, A006880, A086018, A103362.

Denominator[A086018/A006880]

A271798 Decimal expansion of Matthews' constant C_2, an analog of Artin's constant for primitive roots.

Original entry on oeis.org

1, 4, 7, 3, 4, 9, 4, 0, 0, 0, 0, 2, 0, 0, 1, 4, 5, 8, 0, 7, 6, 8, 0, 8, 4, 3, 1, 8, 4, 7, 6, 9, 2, 5, 9, 6, 3, 9, 6, 6, 7, 1, 8, 5, 8, 1, 7, 3, 2, 7, 2, 1, 5, 8, 4, 4, 2, 0, 7, 9, 6, 1, 9, 2, 8, 5, 5, 5, 8, 3, 5, 3, 4, 0, 9, 8, 5, 5, 0, 3, 5, 5, 9, 8, 0, 7, 8, 2, 7, 1, 1, 3, 0, 1, 7, 6, 6, 1, 8, 9, 9, 4, 4, 3, 3, 6

Offset: 0

Jean-François Alcover, Apr 14 2016

			0.147349400002001458076808431847692596396671858173272158442...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Cf. A005596.

digits = 66; m0 = 1000; dm = 100; Clear[s]; r[n_] := RootSum[1 - 2*# - #^2 + #^3& , #^n&] - 1; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], Print[m]; m = m + dm]; RealDigits[s[m]][[1]]

prodeulerrat(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) \\ Amiram Eldar, Mar 16 2021

C_2 = Product_{p prime} 1 - (p^2 - (p - 1)^2)/(p^2*(p - 1)).

Log(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) + O(p,Infinity)^m = Sum_{n=2..m} -r(n)/(n*p^n), where r(n) = rootSum(1 - 2*x - x^2 + x^3, x^n) - 1.

More digits from Vaclav Kotesovec, Jun 19 2020

A321217 Genocchi irregular primes.

Original entry on oeis.org

17, 31, 37, 41, 43, 59, 67, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 149, 151, 157, 193, 223, 229, 233, 241, 251, 257, 263, 271, 277, 281, 283, 293, 307, 311, 313, 331, 337, 347, 353, 379, 389, 397, 401, 409, 421, 431, 433, 439, 449, 457, 461, 463, 467, 491, 499

Offset: 1

Michel Marcus, Oct 31 2018

nonn

An odd prime p is G-irregular if it divides at least one of the integers G2, G4, ..., G(p-3).

Conjecture (Hu et al., 2019): The asymptotic density of this sequence within the primes is 1 - 3*A/(2*sqrt(e)) = 0.659776..., where A is Artin's constant (A005596). - Amiram Eldar, Dec 06 2022

Amiram Eldar, Table of n, a(n) for n = 1..1024
Su Hu, Min-Soo Kim, Pieter Moree, and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, Journal of Number Theory, Vol. 205 (2019), pp. 59-80, preprint, arXiv:1809.08431 [math.NT], 2019.
Peter Luschny, Irregular Bernoulli and Euler Primes.
Pieter Moree and Min Sha, Primes in arithmetic progressions and nonprimitive roots, Bulletin of the Australian Mathematical Society (2019), preprint, arXiv:1901.02650 [math.NT], 2019.
Pieter Moree and Pietro Sgobba, Prime divisors of l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l, arXiv:2209.08047 [math.NT], 2022.

Cf. A036968 (Genocchi numbers), A000928 (irregular primes), A120337 (Euler-irregular primes), A128197 (strong irregular primes), A250216 (weak irregular primes), A005596.

A321217_list := proc(bound)
   local ae, F, p, m, maxp; F := NULL;
   for m from 2 by 2 to bound do
      p := nextprime(m+1);
      ae := abs(m*euler(m-1, 0));
      maxp := min(ae, bound);
      while p <= maxp do
          if ae mod p = 0 then F := F, p fi;
          p := nextprime(p)
      od
   od;
sort({F}) end: A321217_list(500); # Peter Luschny, Nov 11 2018

G[n_] := G[n] = n EulerE[n - 1, 0];
GenocchiIrregularQ[p_] := AnyTrue[Table[G[k], {k, 2, p-3, 2}], Divisible[#, p]&];
Select[Prime[Range[2, 100]], GenocchiIrregularQ] (* Jean-François Alcover, Nov 16 2018 *)

More terms from Peter Luschny, Nov 11 2018

A003969 Inverse Möbius transform of A003959.

Original entry on oeis.org

1, 4, 5, 13, 7, 20, 9, 40, 21, 28, 13, 65, 15, 36, 35, 121, 19, 84, 21, 91, 45, 52, 25, 200, 43, 60, 85, 117, 31, 140, 33, 364, 65, 76, 63, 273, 39, 84, 75, 280, 43, 180, 45, 169, 147, 100, 49, 605, 73, 172, 95, 195, 55, 340, 91, 360, 105, 124, 61, 455, 63, 132, 189

Offset: 1

Marc LeBrun

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003959, A005596, A072691.

f[p_, e_] := ((p + 1)^(e + 1) - 1)/p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, ((f[i,1] + 1)^(f[i,2] + 1) - 1)/f[i,1]); } \\ Amiram Eldar, Oct 23 2022

Multiplicative with a(p^e) = ((p+1)^(e+1)-1)/p. - David W. Wilson, Sep 01 2001

Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691/A005596 = 2.199369... . - Amiram Eldar, Oct 23 2022

More terms from David W. Wilson, Aug 29 2001

A105887 Primes for which -15 is a primitive root.

Original entry on oeis.org

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

N. J. A. Sloane, Apr 24 2005

nonn

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)

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. A005596 (Artin's constant).

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

cp's OEIS Frontend

A007355 Incorrect duplicate of A297408.

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A019353 Primes with primitive root 27.

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Mathematica

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A049229 Primes p such that p-2 is not squarefree.

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A091880 A049232 indexed by A000040.

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Mathematica

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A103362 Reduced numerators of the fraction of primes < 10^n that are full reptend primes.

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A103363 Reduced denominators of the fraction of primes < 10^n that are full reptend primes.

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A271798 Decimal expansion of Matthews' constant C_2, an analog of Artin's constant for primitive roots.

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Mathematica

PARI

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Extensions

A321217 Genocchi irregular primes.

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Maple

Mathematica

Extensions

A003969 Inverse Möbius transform of A003959.