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A105874
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Primes for which -2 is a primitive root.
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4
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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
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1,1
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COMMENTS
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Also primes for which (p-1)/2 (==-1/2 mod p) is a primitive root. [Joerg Arndt, Jun 27 2011]
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LINKS
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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.
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FORMULA
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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
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(PARI) forprime(p=3, 10^4, if(p-1==znorder(Mod(-2, p)), print1(p", "))); /* Joerg Arndt, Jun 27 2011 */
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Apr 24 2005
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STATUS
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approved
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