0,2

In other words, least m such that 2n+1 divides 2^m-1.

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.

Concerning the complexity of computing this sequence, see for example Bach And Shallit, p. 115, exercise 8.

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

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

E. Bach and J. O. Shallit, Algorithmic Number Theory, I.

A. J. C. Cunningham, On Binal Fractions, Math. Gaz., 4 (1908), circa p. 266.

T. Folger, "Shuffling Into Hyperspace," Discover, 1991 (vol 12, no 1), pages 66-67.

M. Gardner, "Card Shuffles," Mathematical Carnival chapter 10, pages 123-138. New York: Vintage Books, 1977.

M. J. Gardner and C. A. McMahan, Riffling casino checks, Math. Mag., 50 (1977), 38-41.

S. W. Golomb, Permutations by cutting and shuffling, SIAM Rev., 3 (1961), 293-297.

V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems [in Russian], Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53.

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).

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 146, Exer. 21.3

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Riffle Shuffle

Eric Weisstein's World of Mathematics, In-Shuffle

Eric Weisstein's World of Mathematics, Out-Shuffle

Eric Weisstein's World of Mathematics, Multiplicative Order

a((3^n-1)/2)=A025192(n) - Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 09 2008

Bisection of A007733: a(n) = A007733(2n+1) [From Max Alekseyev (maxale(AT)gmail.com), Jun 11 2009]

with(numtheory): f := n->order(2, 2*n+1);

(PARI) a(n)=if(n<0, 0, znorder(Mod(2, 2*n+1))) /* Michael Somos Mar 31 2005 */

(MAGMA) [ 1 ] cat [ Modorder(2, 2*n+1): n in [1..72] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Dec 03 2008]

Cf. A070667-A070675, A070676, A053447, A070677, A070681, A070678, A053451, A070679, A070682, A070680, A070683.

Cf. A024222, A006694 (number of cyclotomic cosets), A014664 (order of 2 mod n-th prime).

Sequence in context: A083673 A131388 A131393 this_sequence A064273 A134561 A120947

Adjacent sequences: A002323 A002324 A002325 this_sequence A002327 A002328 A002329

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jan 13, 2000.

More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2003