A105874 - OEIS

%I #37 Mar 31 2024 14:59:14

%S 5,7,13,23,29,37,47,53,61,71,79,101,103,149,167,173,181,191,197,199,

%T 239,263,269,271,293,311,317,349,359,367,373,383,389,421,461,463,479,

%U 487,503,509,541,557,599,607,613,647,653,661,677,701,709,719,743,751,757,773,797

%N Primes for which -2 is a primitive root.

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

%H Joerg Arndt, <a href="/A105874/b105874.txt">Table of n, a(n) for n = 1..10000</a>

%H L. J. Goldstein, <a href="http://www.jstor.org/stable/2316895">Density questions in algebraic number theory</a>, Amer. Math. Monthly, 78 (1971), 342-349.

%H <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>

%F 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

%p 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);

%t pr=-2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* _N. J. A. Sloane_, Jun 01 2010 *)

%t a[p_,q_]:=Sum[2 Cos[2^n Pi/((2 q+1) (2 p+1))], {n,1,2 q p}];

%t Select[Range[400], Reduce[a[#, 1] == 1, Integers] &];

%t 2 % + 1 (* _Gerry Martens_, Apr 28 2015 *)

%o (PARI) forprime(p=3,10^4,if(p-1==znorder(Mod(-2,p)),print1(p", "))); /* _Joerg Arndt_, Jun 27 2011 */

%o (Python)

%o from sympy import n_order, nextprime

%o from itertools import islice

%o def A105874_gen(startvalue=3): # generator of terms >= startvalue

%o     p = max(startvalue-1,2)

%o     while (p:=nextprime(p)):

%o         if n_order(-2,p) == p-1:

%o             yield p

%o A105874_list = list(islice(A105874_gen(),20)) # _Chai Wah Wu_, Aug 11 2023

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Apr 24 2005