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A293213 Primes p with phi(p-1) a primitive root modulo p, where phi(.) is Euler's totient function (A000010). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 15 16:02 EDT 2020. Contains 336505 sequences. (Running on oeis4.)