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 A019567 Order of the Mongean shuffle permutation of 2n cards: a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1. 4
 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, 18, 14, 12, 55, 50, 7, 18, 34, 46, 14, 74, 24, 26, 33, 20, 78, 86, 29, 90, 18, 18, 48, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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. It appears that a(n) = order((-1)^(n+1)*2 in Z_{2n+1}) / f with f=1 when n==2 (mod 3) and for n = 0, 19, 21, 30,33, 52, 55, 61, 63, 70, ..., f=2 else. I don't know how to characterize the "exceptional" n's. - M. F. Hasler, Mar 31 2019 REFERENCES A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134-135. W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, p. 311 LINKS R. J. Mathar, Table of n, a(n) for n = 0..2000 P. Diaconis, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196. Arne Ledet, The Monge shuffle for two-power decks, Math. Scand. Vol 98, No 1 (2006), 5-11. E. Ross, Mathematics and Music: The Mathieu Group M_12 (2011), Chapter 2. T. & X. Vigouroux, First 2000000 terms, for n = 0..1999999 FORMULA a(A163777(n)/2) = A163777(n). - Andrew Howroyd, Nov 11 2017 EXAMPLE Illustrating the initial terms:    n  4n+1  2^m+1  2^m-1  m    0    1            1    1    1    5     5           2    2    9     9           3    3   13    5*13         6    4   17     17          4    5   21           3*21  6    6   25   41*25        10 MAPLE A019567:=  proc(n)     for m from 1 do         if modp(2^m-1, 4*n+1) =0 or modp(2^m+1, 4*n+1)=0 then             return m ;         end if;     end do; end proc: # N. J. A. Sloane, Jul 28 2007 MATHEMATICA a[n_] := For[m=1, True, m++, If[AnyTrue[{-1, 1}, Divisible[2^m+#, 4n+1]&], Return[m]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 26 2019 *) PROG (PARI) A019567(n, z=Mod(2, 4*n+1))=for(m=1, oo, bittest(5, lift(z^m+1))&&return(m)) \\ M. F. Hasler, Mar 31 2019 CROSSREFS Cf. A163777, A238371, A294673. Sequence in context: A306231 A125703 A156688 * A098286 A226615 A138608 Adjacent sequences:  A019564 A019565 A019566 * A019568 A019569 A019570 KEYWORD nonn,easy AUTHOR John Bullitt (metta(AT)world.std.com), N. J. A. Sloane and J. H. Conway EXTENSIONS Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided the Domoryad reference Definition edited by N. J. A. Sloane, Nov 09 2017 STATUS approved

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Last modified February 22 04:10 EST 2020. Contains 332115 sequences. (Running on oeis4.)