A172490 -id:A172490 - cp's OEIS frontend

A118818 Primes p for which there are more primitive roots below p/2 than above p/2.

223, 379, 463, 631, 691, 883, 907, 1051, 1423, 1447, 1543, 1723, 1747, 1783, 1987, 2143, 2179, 2347, 2467, 2591, 2767, 3259, 3307, 3511, 3631, 3691, 3739, 3823, 3907, 4219, 4447, 4507, 4519, 4639, 4987, 5023, 5107, 5119, 5347, 5683, 5923

Offset: 1

Don Reble, Apr 20 2007

nonn

All terms are of the form 4*m + 3 (see A172480 and A172490). - Emmanuel Vantieghem, Nov 07 2016

			223 is a term because it has 38 primitive roots below 111.5, but 34 above 111.5.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A128859, A172480, A172490.

q[n_] := PrimeQ[n] && Length[(p = PrimitiveRootList[n])] > 2*Count[p, ?(# > n/2 &)]; Select[4*Range[0, 1500] + 3, q] (* _Amiram Eldar, Oct 11 2021 *)

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].

Original entry on oeis.org

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

Emmanuel Vantieghem, Feb 04 2010

nonn

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).

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

A002144 is a subsequence.

Cf. A118818, A172490

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

<< 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]

cp's OEIS Frontend

A118818 Primes p for which there are more primitive roots below p/2 than above p/2.

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