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The indices of primes for which the average of the primitive roots is < p/2.
1

%I #20 Jan 09 2016 17:12:38

%S 11,14,19,48,75,94,114,115,117,124,149,153,155,177,182,224,272,300,

%T 324,348,351,365,370,403,465,510,515,522,531,546,555,578,614,634,667,

%U 677,683,707,748,765,788,795,802,808,832,850,871,876,886,888,966,980

%N The indices of primes for which the average of the primitive roots is < p/2.

%C These primes are all of the form p==3 (mod 4). (conjecture)

%e p(a[1])=p(11)=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.

%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, 1000}];Flatten[Position[A, _?(# < 1 &)]]

%Y Cf. A008330, A060749, A088144, A266989.

%K nonn

%O 1,1

%A _Dimitri Papadopoulos_, Jan 08 2016