A267010 Primes of the form p==3 (mod 4) such that the average of their primitive roots equals p/2.

19, 307, 1451, 2179, 2251, 2683, 2843, 3259, 3907, 4447, 11863, 12907, 17623, 30763, 37963, 51059, 52543, 86131, 92467, 104851, 129763, 131203, 146683, 150151, 156151, 156703, 162523, 163819, 174007, 245899, 263827, 287731, 348643, 353611, 400123, 412831, 423091, 432587

Offset: 1

Author

Dimitri Papadopoulos, Jan 08 2016

Keywords

• nonn

Comments

Most primes for which the average of the primitive roots=p/2 are of the form p==1(mod 4). Much rarer for primes of form p==3(mod 4) to have this property. (Observation)

Examples

			19 is a term because the primitive roots of 19 are 2, 3, 10, 13, 14, and 15. Their average is (2+3+10+13+14+15)/phi(18)=57/phi(18)=57/6=19/2.
		

Crossrefs

Cf. A060749. Intersection of A002145 and A266987.

Programs

Maple

isA267010 := proc(n)
    if isprime(n) and modp(n,4) = 3 then
        isA266987(n) ;
    else
        false;
    end if;
end proc: # R. J. Mathar, Aug 14 2024

Mathematica

f[n_] := If[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[n]], {i, Prime[n] - 1}],    Prime[n] - 1]]] == EulerPhi[Prime[n] - 1]*Prime[n]/2, 1, 0];
For[k = 1, k < 10000, k++, If[f[k] == 1 && Mod[Prime[k], 4] == 3, Print[k, "  ", Prime[k]]]]
Select[4*Range[1000] + 3, PrimeQ[#] && Mean[PrimitiveRootList[#]] == #/2 &] (* Amiram Eldar, Oct 11 2021 *)

PARI

vr(p) = j=0; r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); r; \\ after A060749
isok(p) = ((p % 4 == 3) && (vpr = vr(p)) && (vecsum(vpr) == #vpr*p/2)); \\ Michel Marcus, Jan 09 2016

Extensions

a(16)-a(38) from Michel Marcus, Jan 09 2016