A105874 Primes for which -2 is a primitive root.

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

1,1

Comments

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

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A216750 A003628 A216776 * A105904 A038901 A260791

Adjacent sequences: A105871 A105872 A105873 * A105875 A105876 A105877

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2005

Status

approved