A059991 a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).

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

Offset: 1

Author

Thomas Ward, Mar 08 2001

Keywords

• easy
• nonn

Comments

Number of points of period n in the simplest nontrivial disconnected S-integer dynamical system.

This sequence comes from the simplest disconnected S-integer 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_{d|n}mu(d)a(n/d) = 0 mod n for all n and since it comes from a group automorphism it is a divisibility sequence.

Examples

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

Links

• R. Brown and J. L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.
• Vijay Chothi, Periodic Points in S-integer dynamical systems, PhD thesis, University of East Anglia, 1996.
• Vijay Chothi, Graham Everest, Thomas Ward, S-integer dynamical systems: periodic points, J. Reine Angew Math. 489 (1997), 99-132.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• T. Ward, Almost all S-integer dynamical systems have many periodic points, Ergodic Th. Dynam. Sys. 18 (1998), 471-486.
• Index to divisibility sequences

Crossrefs

Cf. A000079, A006519, A129760.

Programs

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^(n-2^ord2)) od:

Mathematica

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

Extensions

More terms from James Sellers, Mar 15 2001