A172480 -id:A172480 - 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 *)

A172490 Primes p of the form 4m+3 for which there are exactly as many primitive roots modulo p in the interval [0,p/2] as in the interval [p/2,p].

Original entry on oeis.org

7, 31, 43, 67, 307, 367, 487, 643, 1327, 1663, 2371, 3643, 3847, 4327, 4951, 6091, 6571, 8263, 9151, 9187, 11239, 11383, 11863, 15307, 24007, 24151, 27847, 30091, 30643, 33619, 36871, 42187, 44171, 46279, 46591, 48787, 70843, 71887, 72103, 72379, 73363, 79867, 82003, 92503, 95467, 106243, 110431, 120943, 126031, 130363, 139759, 143827, 162751, 167107, 173191, 174859, 183247

Offset: 1

Emmanuel Vantieghem, Feb 05 2010

Primes 4*k+3 where half of the primitive roots are <= (p-1)/2.
The sequence is probably infinite.
Primes of the form 4m+1 always have as many primitive roots in [0,p/2] as in [p/2,p] (see A172480).

Cf. A118818, A172480.

with(numtheory): p:=3: while p<1000 do if(p mod 4 = 3)then b1:=0: b2:=0: m:=primroot(p): while not m=FAIL do if(mNathaniel Johnston, Jun 26 2011

<< NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, , 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]

isA172490(p)=isprime(p)&&p%4==3&&sum(n=0,p\2,gcd(n,p)==1&&znorder(Mod(n,p))==p-1)==sum(n=p-p\2,p,gcd(n,p)==1&&znorder(Mod(n,p))==p-1) \\ Charles R Greathouse IV, Jun 27 2011

More terms from Robert Israel, Nov 23 2019

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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