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A059991
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a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).
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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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.
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REFERENCES
| 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.
T. Ward. Almost all S-integer dynamical systems have many periodic points. Ergodic Th. Dynam. Sys. 18 (1998), 471-486.
R. Brown and J. L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.
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LINKS
| Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Chothi's thesis
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EXAMPLE
| a(24) = 2^16 = 65536 because ord_2(24)=3, so 24-2^ord_2(24)=16.
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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:
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MATHEMATICA
| ord2[n_?OddQ] = 0; ord2[n_?EvenQ] := FactorInteger[n][[1, 2]]; a[n_] := 2^(n-2^ord2[n]); a /@ Range[34]
(* From Jean-François Alcover, May 19 2011, after Maple prog. *)
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CROSSREFS
| Cf. A000079, A006519, A129760.
Sequence in context: A117438 A075499 A099394 * A002568 A111661 A072651
Adjacent sequences: A059988 A059989 A059990 * A059992 A059993 A059994
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KEYWORD
| easy,nonn
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AUTHOR
| Thomas Ward (t.ward(AT)uea.ac.uk), Mar 08 2001
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Mar 15 2001
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