cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-27 of 27 results.

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

Views

Author

David W. Wilson

Keywords

Comments

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)

Links

Crossrefs

Cf. A019334, A005596, A000720, A323594.

Programs

  • Mathematica
    pr=27; Select[Prime[Range[300]], MultiplicativeOrder[pr, # ] == #-1 &]
  • PARI
    isA019353(n) = isprime(n) && (n!=3) && znorder(Mod(27,n)) == n-1 \\ Jianing Song, May 12 2024

A241043 Primes having primitive roots 2 and 3.

Original entry on oeis.org

5, 19, 29, 53, 101, 139, 149, 163, 173, 197, 211, 269, 293, 317, 379, 389, 461, 509, 557, 653, 677, 701, 773, 797, 821, 859, 907, 941, 1061, 1109, 1123, 1229, 1277, 1291, 1301, 1373, 1483, 1493, 1637, 1733, 1747, 1901, 1949, 1973, 1987, 1997, 2069, 2083
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[400]], fQ[2, #] && fQ[3, #] &]

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

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A019335, A213052.

Programs

  • Mathematica
    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

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

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

Programs

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

A241048 Primes having primitive roots 2, 3, 5, 7, 11, 13, and 17.

Original entry on oeis.org

2477, 9173, 10853, 13877, 14957, 15413, 22277, 22613, 24317, 27653, 30197, 34877, 37253, 41117, 41333, 42437, 42677, 43973, 48677, 51413, 55733, 61613, 62597, 63773, 66293, 72533, 73757, 74093, 76733, 79397, 79757, 82997, 86357, 90173, 92237, 92333, 95597
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A019334, A019335, A019337, A019339, A019341, A019344, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[10000]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] && fQ[13, #] && fQ[17, #] &]

A071428 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(3).

Original entry on oeis.org

4, 6, 16, 18, 28, 30, 42, 52, 78, 88, 100, 112, 126, 136, 138, 148, 162, 172, 196, 198, 210, 222, 232, 256, 268, 280, 282, 292, 316, 330, 352, 378, 388, 400, 448, 460, 462, 486, 508, 520, 556, 568, 570, 592, 606, 616, 630, 640, 652, 676, 690, 700, 738, 750
Offset: 1

Views

Author

Robert G. Wilson v, Jun 22 2002

Keywords

Comments

n such that n+1 is a prime with primitive root 3 (A019334 except for the first term). [Joerg Arndt, Jul 05 2011]

Crossrefs

Cf. A071642.

Programs

  • PARI
    for(n=2,1000,if(polisirreducible(Mod(1,3)*sum(e=0,n,x^e)),print1(n+1,", "))) /* Joerg Arndt, Jul 05 2011 */
  • PARI
    forprime(p=5,1000,if(znorder(Mod(3,p))==p-1,print1(p-1,", "))) /* much faster */ /* Joerg Arndt, Jul 05 2011 */
Previous Showing 21-27 of 27 results.

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