

A049096


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


12



3, 9, 10, 15, 21, 27, 30, 33, 39, 45, 50, 51, 55, 57, 63, 68, 69, 70, 75, 78, 81, 87, 90, 93, 99, 105, 110, 111, 117, 123, 129, 130, 135, 141, 147, 150, 153, 159, 165, 170, 171, 177, 182, 183, 189, 190, 195, 201, 204, 207, 210, 213, 219, 225, 230, 231, 234, 237, 243
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OFFSET

1,1


COMMENTS

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
That sequence is A072936.  Robert Israel, Nov 20 2015
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
From Robert Israel, Nov 20 2015: (Start)
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".
If n is in the sequence, then so is k*n for any odd k. (End)
The sequence consists of all odd multiples of { 3, 10, 55, 68, 78, 182, 301, 406, 666, ... }.  M. F. Hasler, Mar 06 2018


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, When p^2 divides 2^n + 1


FORMULA

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


EXAMPLE

9 is here because 2^9 + 1 = 513 is divisible by 9.
99 is here because 2^99 + 1 = 3^3*19*67*683*5347*20857*242099935645987 is divisible by 9, i.e. is not squarefree.


MAPLE

remove(n > numtheory:issqrfree(2^n+1), [$1..250]); # Robert Israel, Jul 07 2014


MATHEMATICA

Select[Range[243], !SquareFreeQ[2^# + 1] &] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)


PROG

(PARI) is(n)=!issquarefree(2^n+1) \\ Altug Alkan, Nov 20 2015
(MAGMA) [n: n in [3..220]  not IsSquarefree(2^n+1)]; // Vincenzo Librandi, Mar 08 2018


CROSSREFS

Cf. A001220, A049093, A049094, A049095, A072936, A282269, A282270.
Cf. A086982, which is just the same with base b = 10 instead of b = 2.
Sequence in context: A138923 A324584 A325284 * A272653 A030794 A134073
Adjacent sequences: A049093 A049094 A049095 * A049097 A049098 A049099


KEYWORD

nonn


AUTHOR

Labos Elemer


EXTENSIONS

More terms from James A. Sellers, Dec 16 1999
More terms from Vladeta Jovovic, Apr 12 2002
Missing term 182 added by Rainer Rosenthal, Nov 01 2005


STATUS

approved



