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

Author

Dimitri Papadopoulos, Jan 08 2016

Keywords

• nonn

Comments

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)

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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