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%I #29 Oct 11 2021 18:28:23
%S 19,307,1451,2179,2251,2683,2843,3259,3907,4447,11863,12907,17623,
%T 30763,37963,51059,52543,86131,92467,104851,129763,131203,146683,
%U 150151,156151,156703,162523,163819,174007,245899,263827,287731,348643,353611,400123,412831,423091,432587
%N Primes of the form p==3 (mod 4) such that the average of their primitive roots equals p/2.
%C 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)
%e 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.
%t 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];
%t For[k = 1, k < 10000, k++, If[f[k] == 1 && Mod[Prime[k], 4] == 3, Print[k, " ", Prime[k]]]]
%t Select[4*Range[1000] + 3, PrimeQ[#] && Mean[PrimitiveRootList[#]] == #/2 &] (* _Amiram Eldar_, Oct 11 2021 *)
%o (PARI) 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
%o isok(p) = ((p % 4 == 3) && (vpr = vr(p)) && (vecsum(vpr) == #vpr*p/2)); \\ _Michel Marcus_, Jan 09 2016
%Y Cf. A060749. Intersection of A002145 and A266987.
%K nonn
%O 1,1
%A _Dimitri Papadopoulos_, Jan 08 2016
%E a(16)-a(38) from _Michel Marcus_, Jan 09 2016
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