A049096 - OEIS

%I #76 Sep 08 2022 08:44:58

%S 3,9,10,15,21,27,30,33,39,45,50,51,55,57,63,68,69,70,75,78,81,87,90,

%T 93,99,105,110,111,117,123,129,130,135,141,147,150,153,159,165,170,

%U 171,177,182,183,189,190,195,201,204,207,210,213,219,225,230,231,234,237,243

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

%C 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 That sequence is A072936. - _Robert Israel_, Nov 20 2015

%C 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 From _Robert Israel_, Nov 20 2015: (Start)

%C 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 If n is in the sequence, then so is k*n for any odd k. (End)

%C The sequence consists of all odd multiples of { 3, 10, 55, 68, 78, 182, 301, 406, 666, ... }. - _M. F. Hasler_, Mar 06 2018

%H Robert Israel, <a href="/A049096/b049096.txt">Table of n, a(n) for n = 1..10000</a>

%H Robert Israel, <a href="/A049096/a049096.pdf">When p^2 divides 2^n + 1</a>

%F 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 9 is here because 2^9 + 1 = 513 is divisible by 9.

%e 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 remove(n -> numtheory:-issqrfree(2^n+1), [$1..250]); # _Robert Israel_, Jul 07 2014

%t Select[Range[243], !SquareFreeQ[2^# + 1] &] (* _Vladimir Joseph Stephan Orlovsky_, Mar 18 2011*)

%o (PARI) is(n)=!issquarefree(2^n+1) \\ _Altug Alkan_, Nov 20 2015

%o (Magma) [n: n in [3..220] | not IsSquarefree(2^n+1)]; // _Vincenzo Librandi_, Mar 08 2018

%Y Cf. A001220, A049093, A049094, A049095, A072936, A282269, A282270.

%Y Cf. A086982, which is just the same with base b = 10 instead of b = 2.

%K nonn

%O 1,1

%A _Labos Elemer_

%E More terms from _James A. Sellers_, Dec 16 1999

%E More terms from _Vladeta Jovovic_, Apr 12 2002

%E Missing term 182 added by _Rainer Rosenthal_, Nov 01 2005