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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 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

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