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%I A002326 M0936 N0350
%S A002326 1,2,4,3,6,10,12,4,8,18,6,11,20,18,28,5,10,12,36,12,20,14,12,23,21,8,52,
%T A002326 20,18,58,60,6,12,66,22,35,9,20,30,39,54,82,8,28,11,12,10,36,48,30,100,
%U A002326 51,12,106,36,36,28,44,12,24,110,20,100,7,14,130,18,36,68,138,46,60,28
%N A002326 Multiplicative order of 2 mod 2n+1.
%C A002326 In other words, least m such that 2n+1 divides 2^m-1.
%C A002326 Number of riffle shuffles of 2n+2 cards required to return a deck to 
               initial state. A riffle shuffle replaces a list s(1), s(2), ..., 
               s(m) by s(1), s((i/2)+1), s(2), s((i/2)+2), ... a(1) = 2 because 
               a riffle shuffle of [1, 2, 3, 4] requires 2 iterations [1, 2, 3, 
               4] -> [1, 3, 2, 4] -> [1, 2, 3, 4] to restore the original order.
%C A002326 Concerning the complexity of computing this sequence, see for example 
               Bach And Shallit, p. 115, exercise 8.
%C A002326 It is not difficult to prove that if 2n+1 is a prime then 2n is divisible 
               by a(n). We conjecture that, conversely, if 2n is divisible by a(n) 
               then 2n+1 is 1 or a prime. - Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               Apr 29 2008
%C A002326 It is not difficult to prove that if 2n+1 is a prime then 2n is a multiple 
               of a(n). But the converse is not true. Indeed, one can prove that 
               a(2^(2t-1))=4t. Thus if n=2^(2t-1), where, for any m>0, t=2^(m-1) 
               then 2n is a multiple of a(n) while 2n+1 is a Fermat number which, 
               as is well known, is not always a prime. It is an interesting problem 
               to describe all composite numbers for which 2n is divisible by a(n). 
               - Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 09 2008
%D A002326 E. Bach and J. O. Shallit, Algorithmic Number Theory, I.
%D A002326 A. J. C. Cunningham, On Binal Fractions, Math. Gaz., 4 (1908), circa 
               p. 266.
%D A002326 T. Folger, "Shuffling Into Hyperspace," Discover, 1991 (vol 12, no 1), 
               pages 66-67.
%D A002326 M. Gardner, "Card Shuffles," Mathematical Carnival chapter 10, pages 
               123-138. New York: Vintage Books, 1977.
%D A002326 M. J. Gardner and C. A. McMahan, Riffling casino checks, Math. Mag., 
               50 (1977), 38-41.
%D A002326 S. W. Golomb, Permutations by cutting and shuffling, SIAM Rev., 3 (1961), 
               293-297.
%D A002326 V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems 
               [in Russian], Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53.
%D A002326 L. Lunelli and M. Lunelli, Tavola di congruenza a^n == 1 mod K per a=2,
               5,10, Atti Sem. Mat. Fis. Univ. Modena 10 (1960/61), 219-236 (1961).
%D A002326 J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson 
               Education, Inc, 2006, p. 146, Exer. 21.3
%D A002326 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002326 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002326 T. D. Noe, <a href="b002326.txt">Table of n, a(n) for n = 0..10000</a>
%H A002326 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Haupt-Exponent.html">Link to a section of The World of Mathematics.</
               a>
%H A002326 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiffleShuffle.html">Riffle Shuffle</a>
%H A002326 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               In-Shuffle.html">In-Shuffle</a>
%H A002326 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Out-Shuffle.html">Out-Shuffle</a>
%H A002326 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MultiplicativeOrder.html">Multiplicative Order</a>
%F A002326 a((3^n-1)/2)=A025192(n) - Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               May 09 2008
%F A002326 Bisection of A007733: a(n) = A007733(2n+1) [From Max Alekseyev (maxale(AT)gmail.com), 
               Jun 11 2009]
%p A002326 with(numtheory): f := n->order(2,2*n+1);
%o A002326 (PARI) a(n)=if(n<0,0,znorder(Mod(2,2*n+1))) /* Michael Somos Mar 31 2005 
               */
%o A002326 (MAGMA) [ 1 ] cat [ Modorder(2, 2*n+1): n in [1..72] ]; [From Klaus Brockhaus 
               (klaus-brockhaus(AT)t-online.de), Dec 03 2008]
%Y A002326 Cf. A070667-A070675, A070676, A053447, A070677, A070681, A070678, A053451, 
               A070679, A070682, A070680, A070683.
%Y A002326 Cf. A024222, A006694 (number of cyclotomic cosets), A014664 (order of 
               2 mod n-th prime).
%Y A002326 Sequence in context: A083673 A131388 A131393 this_sequence A064273 A134561 
               A120947
%Y A002326 Adjacent sequences: A002323 A002324 A002325 this_sequence A002327 A002328 
               A002329
%K A002326 nonn,easy,nice
%O A002326 0,2
%A A002326 N. J. A. Sloane (njas(AT)research.att.com).
%E A002326 More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jan 13, 
               2000.
%E A002326 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2003

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