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Primes with 3 as smallest primitive root.
(Formerly M4356 N1825)
11

%I M4356 N1825 #59 Feb 13 2023 11:16:43

%S 7,17,31,43,79,89,113,127,137,199,223,233,257,281,283,331,353,401,449,

%T 463,487,521,569,571,593,607,617,631,641,691,739,751,809,811,823,857,

%U 881,929,953,977,1013,1039,1049,1063,1087,1097,1193,1217

%N Primes with 3 as smallest primitive root.

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.

%D M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 57.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001123/b001123.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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

%t Prime[ Select[ Range[200], PrimitiveRoot[ Prime[ # ]] == 3 & ]]

%t (* or *)

%t Select[ Prime@Range@200, PrimitiveRoot@# == 3 &] (* _Robert G. Wilson v_, May 11 2001 *)

%o (PARI) forprime(p=3, 1000, if(znorder(Mod(2, p))!=p-1&&znorder(Mod(3, p))==p-1, print1(p,", ")));

%o (PARI) { n=0; forprime (p=3, 99999, if (znorder(Mod(2,p))!=p-1 && znorder(Mod(3,p))==p-1, n++; write("b001123.txt", n, " ", p); if (n>=1000, break) ) ) } \\ _Harry J. Smith_, Jun 14 2009

%o (Python)

%o from itertools import islice

%o from sympy import nextprime, is_primitive_root

%o def A001123_gen(): # generator of terms

%o p = 3

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

%o if not is_primitive_root(2,p) and is_primitive_root(3,p):

%o yield p

%o A001123_list = list(islice(A001123_gen(),30)) # _Chai Wah Wu_, Feb 13 2023

%Y Cf. A001122, A001124, etc.

%Y Cf. A019334.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, May 10 2001