

A014657


Numbers m that divide 2^k + 1 for some nonnegative k.


10



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
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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, (m1)(2^e1)/m is a term of A253608. Moreover, e(n) is 2*A195610(n) when m is a(n).  Donald M Davis, Jan 12 2018
From Wolfdieter Lang, Aug 22 2020: (Start)
Without a(2) = 2 this is the complement of A014659 relative to the odd positive integers A005408.
For the least nonnegative integer k(n) with 2^k(n) + 1 == d(n)*a(n), for n >= 1, see k(n) = A195610(n) and d(n) = A337220(n).
Starting with a(3) = 3 these numbers are the odd moduli, named 2*n+1 in the definition of A003558, for which the minus signs applies (see A332433(m) for the signs applying for A003558(m)). (End)


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Moree, Appendix to V. Pless et al., Cyclic SelfDual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 4869, 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] (* JeanFrançois 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 !! (n1)
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, A005408, A014659, A014661, A091317, A179480, A195470, A195610, A332433, A337220.
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



