A135366 a(n) is the smallest nonnegative k such that n divides 2^k + k.

0, 2, 1, 4, 4, 2, 6, 8, 7, 4, 3, 8, 12, 6, 7, 16, 16, 14, 18, 4, 19, 8, 22, 8, 33, 12, 7, 40, 11, 26, 23, 32, 8, 16, 6, 32, 5, 18, 37, 24, 40, 38, 42, 8, 7, 22, 10, 32, 61, 84, 38, 12, 35, 32, 46, 40, 32, 28, 24, 44

Offset: 1

Author

John L. Drost, Feb 16 2008

Keywords

• nonn

Comments

a(2^m) = 2^m for m > 0. If p is an odd prime then by Fermat, a(p) <= p-1. 25 is the smallest n with a(n) > n.

Examples

			a(9)=7 since 2^7 + 7 = 9*15 and 2^k + k is not divisible by 9 for 0 <= k < 7.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• International Mathematical Olympiad, Problem N7, IMO-2006, p. 63.

Crossrefs

Cf. A135359, A247248.

Programs

Mathematica

sk[n_]:=Module[{k=0},While[!Divisible[2^k+k,n],k++];k]; Array[sk,60] (* Harvey P. Dale, Jun 01 2013 *)

PARI

a(n) = for(m=0, oo, if(Mod(2, n)^m==-m, return(m))); \\ Jinyuan Wang, Mar 15 2020

Extensions

Corrected by Harvey P. Dale, Jun 01 2013