A069051 - OEIS

%I #126 Aug 11 2024 14:34:05

%S 2,3,7,19,43,127,163,379,487,883,1459,2647,3079,3943,5419,9199,11827,

%T 14407,16759,18523,24967,26407,37339,39367,42463,71443,77659,95923,

%U 99079,113779,117307,143263,174763,175447,184843,265483,304039,308827

%N Primes p such that p-1 divides 2^p-2.

%C These are the prime values of n such that 2^n == 2 (mod n*(n-1)). - _V. Raman_, Sep 17 2012

%C 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

%C 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

%C From _Thomas Ordowski_, Nov 20 2018: (Start)

%C Conjecture: if n-1 divides 2^n-2, then (2^n-2)/(n-1) is squarefree.

%C 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)

%C 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

%H Charles R Greathouse IV, <a href="/A069051/b069051.txt">Table of n, a(n) for n = 1..3314</a> (First 553 terms from V. Raman)

%H Mersenne Forum, <a href="http://mersenneforum.org/showthread.php?t=17198"> Prime Conjecture </a>

%t Select[Prime[Range[10000]], Mod[2^# - 2, # - 1] == 0 &] (* _T. D. Noe_, Sep 19 2012 *)

%t Join[{2,3},Select[Prime[Range[30000]],PowerMod[2,#,#-1]==2&]] (* _Harvey P. Dale_, Apr 17 2022 *)

%o (PARI) isA069051(p)=Mod(2,p-1)^p==2 && isprime(p); \\ _Charles R Greathouse IV_, Sep 19 2012

%o (Magma) [p : p in PrimesUpTo(310000) | IsZero((2^p-2) mod (p-1))]; // _Vincenzo Librandi_, Dec 03 2018

%o (GAP) Filtered([1..350000],p->IsPrime(p) and (2^p-2) mod (p-1)=0); # _Muniru A Asiru_, Dec 03 2018

%o (Python)

%o from sympy import prime

%o for n in range(1,350000):

%o     if (2**prime(n)-2) % (prime(n)-1)==0:

%o         print(prime(n)) # _Stefano Spezia_, Dec 07 2018

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

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

%K nonn

%O 1,1

%A _Benoit Cloitre_, Apr 03 2002

%E a(1) added by _Charles R Greathouse IV_, Sep 19 2012