A211203 -id:A211203 - cp's OEIS frontend

A069051 Primes p such that p-1 divides 2^p-2.

2, 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367, 42463, 71443, 77659, 95923, 99079, 113779, 117307, 143263, 174763, 175447, 184843, 265483, 304039, 308827

Offset: 1

Benoit Cloitre, Apr 03 2002

nonn

These are the prime values of n such that 2^n == 2 (mod n*(n-1)). - V. Raman, Sep 17 2012
These are the prime values p such that n^(2^(p-1)) is congruent to n or -n (mod p) for all n in Z/pZ, the commutative ring associated with each term. This results follows from Fermat's little theorem. - Philip A. Hoskins, Feb 08 2013
A prime p is in this sequence iff p-1 belongs to A014741. For p>2, this is equivalent to (p-1)/2 belonging to A014945. - Max Alekseyev, Aug 31 2016
From Thomas Ordowski, Nov 20 2018: (Start)
Conjecture: if n-1 divides 2^n-2, then (2^n-2)/(n-1) is squarefree.
Numbers n such that b^n == b (mod (n-1)*n) for every integer b are 2, 3, 7, and 43; i.e., only prime numbers of the form A014117(k) + 1. (End)
These are primes p such that p^2 divides b^(2^p-2) - 1 for every b coprime to p. - Thomas Ordowski, Jul 01 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..3314 (First 553 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A216822, A217468, A211203, A211349, A330382.

a(n)-1 form subsequence of A014741; (a(n)-1)/2 for n>1 forms a subsequence of A014945.

Filtered([1..350000],p->IsPrime(p) and (2^p-2) mod (p-1)=0); # Muniru A Asiru, Dec 03 2018

[p : p in PrimesUpTo(310000) | IsZero((2^p-2) mod (p-1))]; // Vincenzo Librandi, Dec 03 2018

Select[Prime[Range[10000]], Mod[2^# - 2, # - 1] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{2,3},Select[Prime[Range[30000]],PowerMod[2,#,#-1]==2&]] (* Harvey P. Dale, Apr 17 2022 *)

isA069051(p)=Mod(2,p-1)^p==2 && isprime(p); \\ Charles R Greathouse IV, Sep 19 2012

from sympy import prime
for n in range(1,350000):
    if (2**prime(n)-2) % (prime(n)-1)==0:
        print(prime(n)) # Stefano Spezia, Dec 07 2018

a(1) added by Charles R Greathouse IV, Sep 19 2012

A211349 Primes p such that p-1 divides 2^p + 2.

Original entry on oeis.org

2, 3, 11, 251, 5051, 16811, 2025251, 8751251, 16607051, 28257611, 69005051, 78906251, 176775251, 210381251, 372175451, 550427051, 707025251, 854704451, 1866788051, 2441406251, 2605806251, 4249701251, 5469531251, 9304386251, 10315761251, 10915095251

Offset: 1

Philip A. Hoskins, Feb 06 2013

nonn

Prime p>2 is in this sequence iff (p-1)/2 is in A015950. - Max Alekseyev, Dec 26 2017

Max Alekseyev, Table of n, a(n) for n = 1..177

Cf. A069051, A211203.

Select[Prime[Range[1000]], Mod[2^# + 2, # - 1] == 0 &]

N=10^9;
default(primelimit,N);
forprime(p=2,N, if (-2==Mod(2,p-1)^p, print1(p,", ")));
/* Joerg Arndt, Feb 06 2013 */

from sympy import primerange
A211349_list = [p for p in primerange(1,10**6) if p == 2 or pow(2,p,p-1) == p-3] # Chai Wah Wu, Mar 25 2021

a(19)-a(47) from Giovanni Resta, Feb 10 2013

a(48)-a(177) from Max Alekseyev, Jan 06 2018

cp's OEIS Frontend

A069051 Primes p such that p-1 divides 2^p-2.

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