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.

Showing 1-7 of 7 results.

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

Views

Author

N. J. A. Sloane, Apr 24 2005

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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

A038882 Primes that are not in A038881.

Original entry on oeis.org

2, 3, 13, 17, 23, 29, 31, 41, 47, 59, 61, 67, 71, 73, 101, 103, 109, 149, 163, 173, 179, 191, 193, 197, 199, 223, 233, 241, 251, 277, 281, 293, 311, 331, 337, 349, 367, 373, 379, 383, 409, 419, 443, 457, 461, 463
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, only entries p == 1 (mod 4) of the sequence are not squares mod 11 (from the quadratic reciprocity law). - Lekraj Beedassy, Jul 21 2004
Except for 2, inert primes in Z[sqrt(11)]. 2 splits as (-1)*(3 - sqrt(11))*(3 + sqrt(11)). Cf. A296936. - Alonso del Arte, Jan 02 2015

Links

Crossrefs

Cf. A019339, A038881, A296936.

Programs

  • Mathematica
    Select[Prime@Range[120], JacobiSymbol[11, #] == -1 &] (* Vincenzo Librandi, Sep 08 2012 *)
  • PARI
    isok(p) = isprime(p) && !((p%2) && issquare(Mod(11, p))); \\ Michel Marcus, Jul 04 2023

Extensions

Offset changed from 0 to 1 by Vincenzo Librandi, Sep 08 2012
Definition edited by N. J. A. Sloane, Jul 04 2023

A061324 Primes with 11 as smallest positive primitive root.

Original entry on oeis.org

643, 719, 769, 839, 1009, 1129, 1201, 1511, 1597, 1657, 1801, 2399, 2857, 3049, 3191, 3359, 3889, 4079, 4201, 4561, 4591, 4679, 4861, 4871, 4969, 5039, 5351, 5521, 5591, 5689, 5879, 6271, 6719, 8039, 8111, 8231, 8779, 8929, 9551, 9791, 9973
Offset: 1

Views

Author

Klaus Brockhaus, Apr 24 2001

Keywords

Links

Crossrefs

Cf. A001918, A001126, A019339.

Programs

A071566 Numbers k such that x^k + x^(k-1) + x^(k-2) + ... + x + 1 is irreducible over GF(11).

Original entry on oeis.org

2, 12, 16, 22, 28, 30, 40, 46, 58, 66, 70, 72, 100, 102, 108, 148, 162, 172, 178, 196, 222, 232, 250, 276, 280, 292, 330, 366, 372, 382, 418, 442, 460, 462, 466, 486, 498, 556, 568, 586, 592, 598, 600, 612, 618, 642, 646, 672, 676, 682, 700, 718, 760, 768
Offset: 1

Views

Author

Robert G. Wilson v, Jun 22 2002

Keywords

Crossrefs

Cf. A071642.

Formula

a(n) = A019339(n+1) - 1. - Ralf Stephan, Dec 26 2004

A241046 Primes having primitive roots 2, 3, 5, 7, and 11.

Original entry on oeis.org

173, 293, 677, 2083, 2477, 3533, 3797, 4133, 4157, 4373, 4603, 4637, 5477, 5717, 5923, 6173, 7013, 9173, 9533, 9677, 10853, 11587, 12437, 13037, 13397, 13613, 13877, 14717, 14957, 15077, 15413, 16253, 17093, 17573, 17597, 18413, 18773, 18917, 19157, 19997
Offset: 1

Views

Author

T. D. Noe, Apr 16 2014

Keywords

Links

Crossrefs

Cf. A001122, A001123, A001124, A001126, A019339, A213052.

Programs

  • Mathematica
    fQ[p_, n_] := MultiplicativeOrder[p, n] == n - 1; Select[Prime[Range[2300]], fQ[2, #] && fQ[3, #] && fQ[5, #] && fQ[7, #] && fQ[11, #] &]

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, #] &]
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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