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A014657 Numbers m that divide 2^k + 1 for some k. 9
1, 2, 3, 5, 9, 11, 13, 17, 19, 25, 27, 29, 33, 37, 41, 43, 53, 57, 59, 61, 65, 67, 81, 83, 97, 99, 101, 107, 109, 113, 121, 125, 129, 131, 137, 139, 145, 149, 157, 163, 169, 171, 173, 177, 179, 181, 185, 193, 197, 201, 205, 209, 211, 227, 229, 241, 243, 249, 251, 257, 265 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Since for some a < n, 2^a == 1 (mod n) (a consequence of Euler's Theorem), searching up to k=n is sufficient to determine whether an integer is in the sequence. - Michael B. Porter, Dec 06 2009

A195470(a(n)) > 0; A195610(n) gives the smallest k such that a(n) divides 2^k + 1. - Reinhard Zumkeller, Sep 21 2011

This sequence is the subset of odd integers > 1 as (2*n - 1) in A179480, such that the corresponding entry in A179480 is odd. Example: A179480(14) = 5, odd, with (2*14 - 1) = 27; and 5 is a term of this sequence. A014659 (odd and does not divide (2^k + 1) for any k > 1) represents the subset of odd terms >1 corresponding to A179480 entries that are even. - Gary W. Adamson, Aug 20 2012

All prime factors of a(n) are in A091317.  Sequence has asymptotic density 0. - Robert Israel, Aug 12 2014

This sequence, for m>2, is those m for which, for some e, (m-1)(2^e-1)/m is a term of A253608. Moreover, e(n) is 2*A195610(n) when m is a(n). - Donald M Davis, Jan 12 2018

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

MAPLE

select(t -> [msolve(2^x+1, t)] <> [], [2*i+1 $ i=1..1000]); # Robert Israel, Aug 12 2014

MATHEMATICA

ok[n_] := Module[{k=0}, While[k<=n && Mod[2^k + 1, n] > 0, k++]; k<n]; Select[Range[265], ok] (* Jean-Fran├žois Alcover, Apr 06 2011, after PARI prog *)

okQ[n_] := Module[{k = MultiplicativeOrder[2, n]}, EvenQ[k] && Mod[2^(k/2) + 1, n] == 0]; Join[{1, 2}, Select[Range[3, 265, 2], okQ]] (* T. D. Noe, Apr 06 2011 *)

PROG

(PARI) isA014657(n) = {local(r); r=0; for(k=0, n, if(Mod(2^k+1, n)==Mod(0, n), r=1)); r} \\ Michael B. Porter, Dec 06 2009

(Haskell)

import Data.List (findIndices)

a014657 n = a014657_list !! (n-1)

a014657_list = map (+ 1) $ findIndices (> 0) $ map a195470 [1..]

-- Reinhard Zumkeller, Sep 21 2011

CROSSREFS

Besides initial terms 1 and 2, a subsequence of A296243. Their set difference is given by A296244.

Cf. A000051, A003558, A014659, A014661, A091317, A179480, A195470, A195610.

Sequence in context: A191183 A078645 A067139 * A171056 A161514 A215779

Adjacent sequences:  A014654 A014655 A014656 * A014658 A014659 A014660

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Henry Bottomley, May 19 2000

Extended and corrected by David W. Wilson, May 01 2001

STATUS

approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)