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

1,1

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.

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.

Example

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

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}]

Crossrefs

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

Sequence in context: A191030 A233074 A100031 * A215278 A126975 A175444

Adjacent sequences: A293210 A293211 A293212 * A293214 A293215 A293216

Keyword

nonn

Author

Zhi-Wei Sun, Oct 02 2017

Status

approved