%I #16 Mar 31 2024 15:01:23

%S 2,5,11,17,23,29,47,53,59,71,83,89,101,107,113,131,137,149,167,173,

%T 179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359,

%U 383,389,401,419,443,449,461,467,479,503,509,521,557,563,569,587,593,599,617,641

%N Primes for which -3 is a primitive root.

%C 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 Vincenzo Librandi, <a href="/A105875/b105875.txt">Table of n, a(n) for n = 1..1000</a>

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

%t pr=-3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

%o (Python)

%o from sympy import n_order, nextprime

%o from itertools import islice

%o def A105875_gen(startvalue=2): # generator of terms >= startvalue

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

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

%o if p!=3 and n_order(-3,p) == p-1:

%o yield p

%o A105875_list = list(islice(A105875_gen(),20)) # _Chai Wah Wu_, Aug 11 2023

%Y Cf. A105874.

%K nonn

%O 1,1

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