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 A049096 Numbers k such that 2^k + 1 is divisible by a square > 1. 13
 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, !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 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 September 24 17:17 EDT 2022. Contains 356945 sequences. (Running on oeis4.)