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A293213 Primes p with phi(p-1) a primitive root modulo p, where phi(.) is Euler's totient function (A000010).

%I A293213 #13 Oct 07 2017 14:58:14
%S A293213 2,5,23,43,47,67,101,149,167,211,229,263,269,281,349,353,359,383,389,
%T A293213 421,431,449,461,479,499,503,509,521,661,691,709,719,739,743,829,839,
%U A293213 859,863,883,887,907,941,953,971,983,991,1031,1087,1103,1109,1163,1181,1229,1237,1279,1291,1319,1327,1367,1373
%N A293213 Primes p with phi(p-1) a primitive root modulo p, where phi(.) is Euler's totient function (A000010).
%C A293213 It is well known that for any prime p the number of distinct primitive roots modulo p among 1,...,p-1 is phi(p-1).
%C A293213 Conjecture: The sequence contains infinitely many terms. Moreover, the number of primes p <= x with phi(p-1) a primitive root modulo p is asymptotically equivalent to c*x/(log x) as x tends to the infinity, where c is a constant with 0.36 < c < 0.37.
%H A293213 Zhi-Wei Sun, <a href="/A293213/b293213.txt">Table of n, a(n) for n = 1..10000</a>
%H A293213 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv:1405.0290 [math.NT], 2014.
%e A293213 a(2) = 5 since phi(5-1) = 2 is a primitive root modulo the prime 5.
%t A293213 p[n_]:=p[n]=Prime[n];
%t A293213 n=0;Do[Do[If[Mod[EulerPhi[p[k]-1]^(Part[Divisors[p[k]-1],i])-1,p[k]]==0,Goto[aa]],{i,1,Length[Divisors[p[k]-1]]-1}];
%t A293213 n=n+1;Print[n," ",p[k]];Label[aa],{k,1,220}]
%Y A293213 Cf. A000010, A000040, A242748, A242750, A291615, A291657.
%K A293213 nonn
%O A293213 1,1
%A A293213 _Zhi-Wei Sun_, Oct 02 2017