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 A019567 a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1. 1

%I

%S 1,2,3,6,4,6,10,14,5,18,10,12,21,26,9,30,6,22,9,30,27,8,11,10,24,50,

%T 12,18,14,12,55,50,7,18,34,46,14,74,24,26,33,20,78,86,29,90,18,18,48,

%U 98,33,10,45,70,15,24,60,38,29,78,12,84,41,110,8,84,26,134,12,46,35,36,68,146

%N a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1.

%C Write down 1, then 2 to left, 3 to right, 4 to left, ..., getting [ 2n,2n-2,...,4,2,1,3,5,...,2n-1 ]; the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives order of permutation sending 1 to 2n, 2 to 2n-2, ..., 2n to 2n-1.

%C Equivalently, the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives the number of Mongean shuffles needed to return a deck of 2n cards (n=1,2,3,...) to its original order.

%D A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134-135.

%D W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, p. 311.

%H R. J. Mathar, <a href="/A019567/b019567.txt">Table of n, a(n) for n = 0..2000</a>

%e Illustrating the initial terms:

%e n 4n+1 2^m+1 2^m-1 m

%e 0..1...........1...1

%e 1..5.....5.........2

%e 2..9.....9.........3

%e 3.13...5*13........6

%e 4.17.....17........4

%e 5.21..........3*21.6

%e 6.25..41*25.......10

%p A019567:= proc(n)

%p for m from 1 do

%p if modp(2^m-1,4*n+1) =0 or modp(2^m+1,4*n+1)=0 then

%p return m ;

%p end if;

%p end do;

%p end proc: # _N. J. A. Sloane_, Jul 28 2007

%K nonn,easy

%O 0,2

%A John Bullitt (metta(AT)world.std.com), _N. J. A. Sloane_ and J. H. Conway (conway(AT)math.princeton.edu)

%E Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided the Domoryad reference.

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