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A105875 Primes for which -3 is a primitive root.

%I A105875 #16 Mar 31 2024 15:01:23
%S A105875 2,5,11,17,23,29,47,53,59,71,83,89,101,107,113,131,137,149,167,173,
%T A105875 179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359,
%U A105875 383,389,401,419,443,449,461,467,479,503,509,521,557,563,569,587,593,599,617,641
%N A105875 Primes for which -3 is a primitive root.
%C A105875 Also, primes for which -27 is a primitive root. Proof: -27 = (-3)^3, so -27 is a primitive root just when -3 is a primitive root and the prime is not 3k+1. Now if -3 is a primitive root, then -3 is not a quadratic residue and so the prime is not 3k+1. - _Don Reble_, Sep 15 2007
%H A105875 Vincenzo Librandi, <a href="/A105875/b105875.txt">Table of n, a(n) for n = 1..1000</a>
%H A105875 <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>
%t A105875 pr=-3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
%o A105875 (Python)
%o A105875 from sympy import n_order, nextprime
%o A105875 from itertools import islice
%o A105875 def A105875_gen(startvalue=2): # generator of terms >= startvalue
%o A105875     p = max(startvalue-1,1)
%o A105875     while (p:=nextprime(p)):
%o A105875         if p!=3 and n_order(-3,p) == p-1:
%o A105875             yield p
%o A105875 A105875_list = list(islice(A105875_gen(),20)) # _Chai Wah Wu_, Aug 11 2023
%Y A105875 Cf. A105874.
%K A105875 nonn
%O A105875 1,1
%A A105875 _N. J. A. Sloane_, Apr 24 2005