A337119 Primes p such that b^(p-1) == 1 (mod p-1) for all b coprime to p-1.

2, 3, 5, 7, 13, 17, 19, 37, 41, 43, 61, 73, 97, 101, 109, 127, 157, 163, 181, 193, 241, 257, 313, 337, 379, 401, 421, 433, 487, 541, 577, 601, 641, 661, 673, 757, 769, 881, 883, 937, 1009, 1093, 1153, 1201, 1249, 1297, 1321, 1361, 1459, 1601, 1621, 1801, 1861, 1873, 2017, 2029, 2053, 2161, 2269, 2341, 2437, 2521, 2593

Offset: 1

Author

Francois R. Grieu, Aug 17 2020

Keywords

• nonn

Comments

Equivalently: primes p to p-1 a Novák-Carmichael number A124240.

These p are such that for all x in [0,p), and all b coprime to p-1, x^(b^(p-1)) == x (mod p), this follows from the FLT.

Equivalently, primes p such that for all primes q | p-1, q-1 | p-1. Primes such that p-1 is in A124240. No prime of the form 12k+11 is in this sequence. - Paul Vanderveen, Apr 02 2022

Primes p such that B^(b^(p-1)-1) == 1 (mod p^2) for every B coprime to p and for every b coprime to (p-1)*p. - Thomas Ordowski, Sep 01 2024

Examples

			7 is in the sequence because it is prime, 1 and 5 are the integers (mod 6) coprime to 6; 1^6 mod 6 = 1; and 5^6 mod 6 = 1.
11 is not in the sequence because 3 is coprime to 10; and 3^10 mod 10 = 9 <> 1.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..300 from Harvey P. Dale)

Crossrefs

Cf. A124240.

Programs

Mathematica

a={}; For[p=2,p<2600, p=NextPrime[p],b=p-1; While[--b>0&&(GCD[b,p-1]!=1||PowerMod[b,p-1,p-1]==1)];If[b==0,AppendTo[a,p]]];a
bcpQ[n_]:=Module[{b=Select[Range[n-2],CoprimeQ[n-1,#]&]},AllTrue[ b,PowerMod[ #,n-1,n-1]==1&]]; Select[Prime[Range[400]],bcpQ] (* Harvey P. Dale, Jan 01 2022 *)

Python

from math import gcd
from sympy import isprime
def ok(n):
    if not isprime(n): return False
    return all(pow(b, n-1, n-1) == 1 for b in range(2, n) if gcd(b, n-1)==1)
print([k for k in range(2594) if ok(k)]) # Michael S. Branicky, Apr 02 2022