%I A019567
%S A019567 1,2,3,6,4,6,10,14,5,18,10,12,21,26,9,30,6,22,9,30,27,8,11,10,24,50,12,
%T A019567 18,14,12,55,50,7,18,34,46,14,74,24,26,33,20,78,86,29,90,18,18,48,98,33,
%U A019567 10,45,70,15,24,60,38,29,78,12,84,41,110,8,84,26,134,12,46,35,36,68,146
%N A019567 a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible 
               by 4n + 1.
%C A019567 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 A019567 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 A019567 A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; 
               see pp. 134-135.
%D A019567 W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, 
               p. 311.
%e A019567 Illustrating the initial terms:
%e A019567 n 4n+1 2^m+1 2^m-1 m
%e A019567 0..1...........1...1
%e A019567 1..5.....5.........2
%e A019567 2..9.....9.........3
%e A019567 3.13...5*13........6
%e A019567 4.17.....17........4
%e A019567 5.21..........3*21.6
%e A019567 6.25..41*25.......10
%p A019567 (Crude Maple program from N. J. A. Sloane (njas(AT)research.att.com), 
               Jul 28 2007)
%p A019567 f:=proc(n) local m;
%p A019567 for m from 1 to 500 do
%p A019567 if 2^m-1 mod (4*n+1) = 0 then RETURN(m); fi;
%p A019567 if 2^m+1 mod (4*n+1) = 0 then RETURN(m); fi;
%p A019567 od:
%p A019567 -1;
%p A019567 end;
%Y A019567 Sequence in context: A072637 A125703 A156688 this_sequence A098286 A138608 
               A092283
%Y A019567 Adjacent sequences: A019564 A019565 A019566 this_sequence A019568 A019569 
               A019570
%K A019567 nonn,easy
%O A019567 0,2
%A A019567 John Bullitt (metta(AT)world.std.com), N. J. A. Sloane (njas(AT)research.att.com) 
               and J. H. Conway (conway(AT)math.princeton.edu)
%E A019567 Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided 
               the Domoryad reference.