cp's OEIS Frontend

A049096 Numbers k such that 2^k + 1 is divisible by a square > 1.

%I A049096 #77 Jul 02 2025 16:01:57
%S A049096 3,9,10,15,21,27,30,33,39,45,50,51,55,57,63,68,69,70,75,78,81,87,90,
%T A049096 93,99,105,110,111,117,123,129,130,135,141,147,150,153,159,165,170,
%U A049096 171,177,182,183,189,190,195,201,204,207,210,213,219,225,230,231,234,237,243
%N A049096 Numbers k such that 2^k + 1 is divisible by a square > 1.
%C A049096 Conjecture: lim n -> infinity a(n)/n = C exists and 4 < C < 9/2. There seems to be a sequence of primes p such that p^2 never divides numbers of the form 2^x + 1: the first few are 2, 7, 23, 31. - _Benoit Cloitre_, Aug 20 2002
%C A049096 That sequence is A072936. - _Robert Israel_, Nov 20 2015
%C A049096 The first case where 2^n + 1 is divisible by a square that is coprime to n is n = 182 (where 2^182 + 1 is divisible by 1093^2). - _Robert Israel_, Jul 07 2014
%C A049096 From _Robert Israel_, Nov 20 2015: (Start)
%C A049096 Numbers n such that gcd(n, 2^n + 1) > 1 or n = k m where k is odd and 2 m is the order of 2 modulo a Wieferich prime.  See link "When p^2 divides 2^n + 1".
%C A049096 If n is in the sequence, then so is k*n for any odd k. (End)
%C A049096 The sequence consists of all odd multiples of { 3, 10, 55, 68, 78, 182, 301, 406, 666, ... }. - _M. F. Hasler_, Mar 06 2018
%H A049096 Robert Israel, <a href="/A049096/b049096.txt">Table of n, a(n) for n = 1..10000</a>
%H A049096 Robert Israel, <a href="/A049096/a049096.pdf">When p^2 divides 2^n + 1</a>
%F A049096 For any a(n+1) - a(n) <= 6 since numbers of form 3^a*(2k+1) a > 0, k >= 0, are in the sequence (2^(3*(2k+1) + 1 is divisible by 9). So are numbers of the form 20k + 10 since 2^(20k+10) + 1 is divisible by 25, 110k + 55 since 2^(110k+55) + 1 is divisible by 11^2, 78 + 156k since 2^(156k+78) + 1 is divisible by 13^2 ... - _Benoit Cloitre_, Aug 20 2002
%e A049096 9 is here because 2^9 + 1 = 513 is divisible by 9.
%e A049096 99 is here because 2^99 + 1 = 3^3*19*67*683*5347*20857*242099935645987 is divisible by 9, i.e. is not squarefree.
%p A049096 remove(n -> numtheory:-issqrfree(2^n+1), [$1..250]); # _Robert Israel_, Jul 07 2014
%t A049096 Select[Range[243], !SquareFreeQ[2^# + 1] &] (* _Vladimir Joseph Stephan Orlovsky_, Mar 18 2011*)
%o A049096 (PARI) is(n)=!issquarefree(2^n+1) \\ _Altug Alkan_, Nov 20 2015
%o A049096 (Magma) [n: n in [3..220] | not IsSquarefree(2^n+1)]; // _Vincenzo Librandi_, Mar 08 2018
%Y A049096 Cf. A001220, A049093, A049094, A049095, A072936, A282269, A282270.
%Y A049096 Cf. A086982, which is just the same with base b = 10 instead of b = 2.
%K A049096 nonn
%O A049096 1,1
%A A049096 _Labos Elemer_
%E A049096 More terms from _James Sellers_, Dec 16 1999
%E A049096 More terms from _Vladeta Jovovic_, Apr 12 2002
%E A049096 Missing term 182 added by _Rainer Rosenthal_, Nov 01 2005