A014657 Numbers m that divide 2^k + 1 for some nonnegative k.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

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 Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

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.

Programs

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

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++]; kJean-François 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 *)

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

Extensions

More terms from Henry Bottomley, May 19 2000

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