

A059991


a(n) = 2^(n2^ord_2(n)) (or 2^(nA006519(n))).


3



1, 1, 4, 1, 16, 16, 64, 1, 256, 256, 1024, 256, 4096, 4096, 16384, 1, 65536, 65536, 262144, 65536, 1048576, 1048576, 4194304, 65536, 16777216, 16777216, 67108864, 16777216, 268435456, 268435456, 1073741824, 1, 4294967296, 4294967296
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OFFSET

1,3


COMMENTS

Number of points of period n in the simplest nontrivial disconnected Sinteger dynamical system.
This sequence comes from the simplest disconnected Sinteger system that is not hyperbolic. In the terminology of the papers referred to, it is constructed by choosing the under lying field to be F_2(t), the element to be t and the nontrivial valuation to correspond to the polynomial 1+t. Since it counts periodic points, it satisfies the nontrivial congruence sum_{dn}mu(d)a(n/d) = 0 mod n for all n and since it comes from a group automorphism it is a divisibility sequence.


LINKS

Table of n, a(n) for n=1..34.
R. Brown and J. L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115121.
Vijay Chothi, Periodic Points in Sinteger dynamical systems, PhD thesis, University of East Anglia, 1996. [Broken link]
Vijay Chothi, Graham Everest, Thomas Ward, Sinteger dynamical systems: periodic points, J. Reine Angew Math. 489 (1997), 99132.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Almost all Sinteger dynamical systems have many periodic points, Ergodic Th. Dynam. Sys. 18 (1998), 471486.
Index to divisibility sequences


EXAMPLE

a(24) = 2^16 = 65536 because ord_2(24)=3, so 242^ord_2(24)=16.


MAPLE

readlib(ifactors): for n from 1 to 100 do if n mod 2 = 1 then ord2 := 0 else ord2 := ifactors(n)[2][1][2] fi: printf(`%d, `, 2^(n2^ord2)) od:


MATHEMATICA

ord2[n_?OddQ] = 0; ord2[n_?EvenQ] := FactorInteger[n][[1, 2]]; a[n_] := 2^(n2^ord2[n]); a /@ Range[34]
(* JeanFrançois Alcover, May 19 2011, after Maple prog. *)


CROSSREFS

Cf. A000079, A006519, A129760.
Sequence in context: A075499 A099394 A269698 * A002568 A334063 A111661
Adjacent sequences: A059988 A059989 A059990 * A059992 A059993 A059994


KEYWORD

easy,nonn


AUTHOR

Thomas Ward (t.ward(AT)uea.ac.uk), Mar 08 2001


EXTENSIONS

More terms from James A. Sellers, Mar 15 2001


STATUS

approved



