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

2, 5, 23, 43, 47, 67, 101, 149, 167, 211, 229, 263, 269, 281, 349, 353, 359, 383, 389, 421, 431, 449, 461, 479, 499, 503, 509, 521, 661, 691, 709, 719, 739, 743, 829, 839, 859, 863, 883, 887, 907, 941, 953, 971, 983, 991, 1031, 1087, 1103, 1109, 1163, 1181, 1229, 1237, 1279, 1291, 1319, 1327, 1367, 1373

Offset: 1

Author

Zhi-Wei Sun, Oct 02 2017

Keywords

• nonn

Comments

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).

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.

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Crossrefs

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

Programs

Mathematica

p[n_]:=p[n]=Prime[n];
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}];
n=n+1;Print[n," ",p[k]];Label[aa],{k,1,220}]