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

1,1

Comments

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. How to prove that this set is complete? (End)

Links

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

Mersenne Forum, Prime Conjecture

Mathematica

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 *)

Prog

(PARI) isA069051(p)=Mod(2, p-1)^p==2 && isprime(p); \\ Charles R Greathouse IV, Sep 19 2012
(Magma) [p : p in PrimesUpTo(310000) | IsZero((2^p-2) mod (p-1))]; // Vincenzo Librandi, Dec 03 2018
(GAP) Filtered([1..350000], p->IsPrime(p) and (2^p-2) mod (p-1)=0); # Muniru A Asiru, Dec 03 2018
(Python)
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

Crossrefs

Cf. A216822, A217468, A211203, A211349.

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

Sequence in context: A235616 A257551 A091410 * A229290 A246033 A122724

Adjacent sequences: A069048 A069049 A069050 * A069052 A069053 A069054

Keyword

nonn

Author

Benoit Cloitre, Apr 03 2002

Extensions

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

Status

approved