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A049096 Numbers k such that 2^k + 1 is divisible by a square > 1. 11
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 23 04:16 EST 2020. Contains 331168 sequences. (Running on oeis4.)