A167791 Numbers with primitive root 2.

3, 5, 9, 11, 13, 19, 25, 27, 29, 37, 53, 59, 61, 67, 81, 83, 101, 107, 121, 125, 131, 139, 149, 163, 169, 173, 179, 181, 197, 211, 227, 243, 269, 293, 317, 347, 349, 361, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619

Offset: 1

Author

T. D. Noe, Nov 12 2009

Keywords

• nonn

Comments

Numbers k such that the binary expansion of 1/k has period phi(k). For example 1/27 has a period of 18 bits.

All entries are odd. An odd composite number n can have a primitive root if and only if it is a prime power (see A033948). - V. Raman, Oct 04 2012

It is unknown whether there is a prime p such that p is in this sequence while p^2 is not. - Jianing Song, Jan 27 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Crossrefs

Cf. A000010, A001122 (primes with primitive root 2), A033948.

Programs

Magma

[n: n in [3..619] | IsPrimitive(2, n)]; // Arkadiusz Wesolowski, Dec 22 2020

Mathematica

pr=2; Select[Range[2,2000], MultiplicativeOrder[pr,# ] == EulerPhi[ # ] &]

PARI

for(n=3,200,if(n%2==1&&znorder(Mod(2,n))==eulerphi(n),printf(n","))) \\ V. Raman, Oct 04 2012

PARI

is(n)=n%2 && isprimepower(n) && znorder(Mod(2,n))==eulerphi(n-1) \\ Charles R Greathouse IV, Jul 05 2013