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A135359 a(n) is the smallest nonnegative number k such that n divides 2^k-k.

%I A135359 #14 Sep 08 2022 08:45:32
%S A135359 0,2,4,4,3,4,11,8,5,14,7,4,10,16,16,16,30,16,30,16,11,58,75,16,34,10,
%T A135359 5,16,6,16,8,32,58,30,16,16,58,30,10,16,33,16,54,92,16,118,224,16,36,
%U A135359 34,59,16,36,34,63,16,130,6,64,16,43,8,16,64,16,58,210,84,118,16,43,16,32
%N A135359 a(n) is the smallest nonnegative number k such that n divides 2^k-k.
%H A135359 G. C. Greubel, <a href="/A135359/b135359.txt">Table of n, a(n) for n = 1..1000</a>
%e A135359 a(7)=11, since 2^11-11= 3*7*97 and 2^k-k is not divisible by 7 for 0<=k<11.
%t A135359 b[n_] := Module[{k = 0}, While[! Divisible[2^k - k, n], k++]; k]; Array[b, 25] (* _G. C. Greubel_, Oct 11 2016 *)
%o A135359 (PARI) a(n) = {my(k = 0); while ((2^k-k) % n, k++); k;} \\ _Michel Marcus_, Aug 18 2013
%o A135359 (Magma)
%o A135359 S:=[0];
%o A135359 k:=1;
%o A135359 for n in [2..80] do
%o A135359   while not IsZero((2^k-k) mod n) do
%o A135359        k:=k+1;
%o A135359   end while;
%o A135359   Append(~S, k);
%o A135359   k:=1;
%o A135359 end for;
%o A135359 S; // _Bruno Berselli_, Aug 18 2013
%Y A135359 See A072872 for another version.
%K A135359 nonn
%O A135359 1,2
%A A135359 _John L. Drost_, Feb 16 2008
%E A135359 Edited by _N. J. A. Sloane_, May 27 2010