A266987 -id:A266987 - cp's OEIS frontend

A266986 The indices of primes p for which the average of the primitive roots equals p/2.

1, 3, 6, 7, 8, 10, 12, 13, 16, 18, 21, 24, 25, 26, 29, 30, 33, 35, 37, 40, 42, 44, 45, 50, 51, 53, 55, 57, 59, 60, 62, 63, 65, 66, 68, 70, 71, 74, 77, 78, 79, 80, 82, 84, 87, 88, 89, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 116, 119, 121, 122, 123, 126, 127, 130, 134, 135, 136, 137, 139, 140, 142, 145

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

The average of the primitive roots of a prime p are <,=, or > p/2 (observation).
The indices of all primes p==1(mod 4) are in this sequence since for primes of form 4k+1 b a primitive root implies -b a primitive root.
The indices of some primes p==3 (mod 4) are also in this sequence although for most such primes the average of the primitive roots is <> p/2.(observation)

			p(a(1))=p(1)=2. 2 has the primitive root 1. The average primitive root is 1 and 1=2/2.
p(a(2))=p(3)=5. The primitive roots of 5 are 2 and 3. Their average equals (2+3)/phi(4)=5/2=p/2.

Dimitri Papadopoulos, Table of n, a(n) for n = 1..505

Cf. A008330, A060749, A088144, A266987.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Flatten[Position[A, _?(# == 1 &)]]

A266989 Primes for which the average of the primitive roots is < p/2.

Original entry on oeis.org

31, 43, 67, 223, 379, 491, 619, 631, 643, 683, 859, 883, 907, 1051, 1091, 1423, 1747, 1987, 2143, 2347, 2371, 2467, 2531, 2767, 3307, 3643, 3691, 3739, 3823, 3931, 4019, 4219, 4519, 4691, 4987, 5059, 5107, 5347, 5683, 5827, 6043

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

These primes are congruent to 3 (mod 4).

			a(1)=31. The primitive roots of 31 are 3, 11, 12, 13, 17, 21, 22, and 24.
Their average is (3+11+12+13+17+21+22+24)/phi(30)=123/8<31/2.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A008330, A060749, A088144, A266987, A266988, A267010.

f:= proc(p) local g;
  if not isprime(p) then return false fi;
  g:= numtheory[primroot](p);
  evalb(add(g&^i mod p, i = select(t->igcd(t,p-1)=1, [$1..p-2]))
     < p/2 * numtheory:-phi(p-1))
end proc:
select(f, [seq(i,i=3..10000,4)]); # Robert Israel, Feb 09 2016

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1,100}]; Prime[Flatten[Position[A, _?(# < 1 &)]]]

ar(p) = my(r, pr, j); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsort(r) ;
isok(p) = my(vr = ar(p)); vecsum(vr)/#vr < p/2;
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Feb 09 2016

a(n) = prime(A266988(n)).

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

Original entry on oeis.org

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

Dimitri Papadopoulos, Jan 08 2016

nonn

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)

			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.

Cf. A060749. Intersection of A002145 and A266987.

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

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 *)

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

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

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A266986 The indices of primes p for which the average of the primitive roots equals p/2.

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A266989 Primes for which the average of the primitive roots is < p/2.

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

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