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A105874 Primes for which -2 is a primitive root. 6
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 (list; graph; refs; listen; history; text; internal format)
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
L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
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
Sequence in context: A216750 A003628 A216776 * A105904 A038901 A260791
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 24 2005
STATUS
approved

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Last modified June 18 02:11 EDT 2024. Contains 373468 sequences. (Running on oeis4.)