A105874 - OEIS

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

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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)