A046145 Smallest primitive root modulo n, or 0 if no root exists.

0, 0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 5, 3, 0, 0

Offset

0,4

Comments

The value 0 at index 0 says 0 has no primitive roots, but the 0 at index 1 says 1 has a primitive root of 0, the only real 0 in the sequence.

a(n) is nonzero if and only if n is 2, 4, or of the form p^k, or 2*p^k where p is an odd prime and k>0. - Tom Edgar, Jun 02 2014

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Pēteris K. Siliņš, Cross ratios for finite field geometries, Bachelor's Thesis, Univ. Groningen (Netherlands, 2024). See p. 17.

Eric Weisstein's World of Mathematics, Primitive Root.

Maple

A046145 := proc(n)
  if n <=1 then
    0;
  else
    pr := numtheory[primroot](n) ;
    if pr = FAIL then
       return 0 ;
    else
       return pr ;
    end if;
  end if;
end proc:
seq(A046145(n), n=0..110) ;  # R. J. Mathar, Jul 08 2010

Mathematica

smallestPrimitiveRoot[n_ /; n <= 1] = 0; smallestPrimitiveRoot[n_] := Block[{pr = PrimitiveRoot[n], g}, If[! NumericQ[pr], g = 0, g = 1; While[g <= pr, If[ CoprimeQ[g, n] && MultiplicativeOrder[g, n] == EulerPhi[n], Break[]]; g++]]; g]; smallestPrimitiveRoot /@ Range[0, 100] (* Jean-François Alcover, Feb 15 2012 *)
f[n_] := Block[{pr = PrimitiveRootList[n]}, If[pr == {}, 0, pr[[1]]]]; Array[f, 105, 0] (* v10.0 Robert G. Wilson v, Nov 04 2014 *)

Prog

(PARI) { A046145(n) = for(q=1, n-1, if(gcd(q, n)==1 && znorder(Mod(q, n))==eulerphi(n), return(q); )); 0; } /* V. Raman, Nov 22 2012, edited by Max Alekseyev, Apr 20 2017 */
(Perl) use ntheory ":all"; say "$_ ", znprimroot($_) || 0  for 0..100; # Dana Jacobsen, Mar 16 2017

Crossrefs

Cf. A001918, A046144, A046146, A002233, A071894, A219027, A008330, A010554, A111076, A285512, A285513, A285514.

Sequence in context: A118176 A005731 A132962 * A103309 A248207 A174621

Adjacent sequences: A046142 A046143 A046144 * A046146 A046147 A046148

Keyword

nonn,easy,nice

Author

Eric W. Weisstein

Extensions

Initial terms corrected by Harry J. Smith, Jan 27 2005

Status

approved