A293213 - OEIS

%I #13 Oct 07 2017 14:58:14

%S 2,5,23,43,47,67,101,149,167,211,229,263,269,281,349,353,359,383,389,

%T 421,431,449,461,479,499,503,509,521,661,691,709,719,739,743,829,839,

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

%C 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 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 Zhi-Wei Sun, <a href="/A293213/b293213.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(2) = 5 since phi(5-1) = 2 is a primitive root modulo the prime 5.

%t p[n_]:=p[n]=Prime[n];

%t 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 n=n+1;Print[n," ",p[k]];Label[aa],{k,1,220}]

%Y Cf. A000010, A000040, A242748, A242750, A291615, A291657.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Oct 02 2017