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%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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