A172480 Odd primes p such that there are as many primitive roots (mod p) in the interval [0,p/2] as in the interval [p/2,p].

5, 7, 13, 17, 29, 31, 37, 41, 43, 53, 61, 67, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 307, 313, 317, 337, 349, 353, 367, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487, 509, 521, 541, 557, 569

Offset: 1

Author

Emmanuel Vantieghem, Feb 04 2010

Keywords

• nonn

Comments

The sequence contains all the primes of the form 4m+1 (A002144).

The sequence also contains some primes of the form 4m+3 (see them in A172490).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

A002144 is a subsequence.

Cf. A118818, A172490

Programs

Maple

filter:= proc(p) local m; uses NumberTheory;
  if not isprime(p) then return false fi;
  if p mod 4 = 1 then return true fi;
  m:= Totient(Totient(p))/2;
  PrimitiveRoot(p,ith=m+1)=PrimitiveRoot(p,greaterthan=floor(p/2))
end proc:
select(filter, [seq(i,i=5..1000,2)]); # Robert Israel, Nov 23 2019

Mathematica

<< NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, s = {s, p}, q = (p - 1)/2; g = PrimitiveRoot[p]; se = Select[Range[p - 1], GCD[ #, p - 1] == 1 &]; e = Length[se]; j = 0; t = 0; While[j < e, j++; h = PowerMod[g, se[[j]], p]; If[h <= q, t = t + 1,] ]; If[e == 2t, s = {s, p},] ] ]; s = Flatten[s]