%I #27 Feb 09 2016 20:50:02
%S 31,43,67,223,379,491,619,631,643,683,859,883,907,1051,1091,1423,1747,
%T 1987,2143,2347,2371,2467,2531,2767,3307,3643,3691,3739,3823,3931,
%U 4019,4219,4519,4691,4987,5059,5107,5347,5683,5827,6043
%N Primes for which the average of the primitive roots is < p/2.
%C These primes are congruent to 3 (mod 4).
%H Robert Israel, <a href="/A266989/b266989.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = prime(A266988(n)).
%e a(1)=31. The primitive roots of 31 are 3, 11, 12, 13, 17, 21, 22, and 24.
%e Their average is (3+11+12+13+17+21+22+24)/phi(30)=123/8<31/2.
%p f:= proc(p) local g;
%p if not isprime(p) then return false fi;
%p g:= numtheory[primroot](p);
%p evalb(add(g&^i mod p, i = select(t->igcd(t,p-1)=1, [$1..p-2]))
%p < p/2 * numtheory:-phi(p-1))
%p end proc:
%p select(f, [seq(i,i=3..10000,4)]); # _Robert Israel_, Feb 09 2016
%t 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 &)]]]
%o (PARI) 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) ;
%o isok(p) = my(vr = ar(p)); vecsum(vr)/#vr < p/2;
%o lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ _Michel Marcus_, Feb 09 2016
%Y Cf. A008330, A060749, A088144, A266987, A266988, A267010.
%K nonn
%O 1,1
%A _Dimitri Papadopoulos_, Jan 08 2016