A267010 -id:A267010 - cp's OEIS frontend

A266987 Primes p for which the average of the primitive roots equals p/2.

2, 5, 13, 17, 19, 29, 37, 41, 53, 61, 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, 373, 389, 397, 401, 409, 421, 433, 449

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

From Robert Israel, Feb 01 2016: (Start)
Union of A002144 and A267010.
Contains A002144 because for each of these primes, x is a primitive root iff p-x is a primitive root. (End)

			a(13) = 13 since the primitive roots of 13 are 2, 6, 7, and 11 and the average of these primitive roots is (2+6+7+11)/phi(12) = 26/4 = 13/2.

Robert Israel, Table of n, a(n) for n = 1..8213

Cf. A002144, A008330, A060749, A088144, A266986, A267010.

proots := proc(n)
    local r,eulphi,m;
    if n = 1 then
        return {0} ;
    end if;
    eulphi := numtheory[phi](n) ;
    r := {} ;
    for m from 0 to n-1 do
        if numtheory[order](m,n) = eulphi then
            r := r union {m} ;
        end if;
    end do:
    return r;
end proc:
isA266987 := proc(n)
    local r;
    if isprime(n) then
        r := convert(proots(n),list) ;
        2*add(pr, pr=r)  = n*nops(r) ;
    else
        false;
    end if;
end proc:
for n from 1 to 500 do
    if isA266987(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016
Filter:= proc(p) local x, s,js;
  if p mod 4 = 1 then return true fi;
  x:= numtheory:-primroot(p);
  js:= select(t -> igcd(t,p-1)=1, [$1..p-2]);
  s:= add(x&^ j mod p, j=js);
  evalb(s = p/2 * nops(js))
end proc:
select(Filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 01 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 &)]]]
(* second program (version >= 10): *)
Select[Prime[Range[100]], Mean[PrimitiveRootList[#]] == #/2&] (* Jean-François Alcover, Jan 12 2016 *)

a(n) = prime(A266986(n)).

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)).

cp's OEIS Frontend

A266987 Primes p for which the average of the primitive roots equals p/2.

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