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A003558
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Least number m such that 2^m = +- 1 mod 2n + 1.
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12
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0, 1, 2, 3, 3, 5, 6, 4, 4, 9, 6, 11, 10, 9, 14, 5, 5, 12, 18, 12, 10, 7, 12, 23, 21, 8, 26, 20, 9, 29, 30, 6, 6, 33, 22, 35, 9, 20, 30, 39, 27, 41, 8, 28, 11, 12, 10, 36, 24, 15, 50, 51, 12, 53, 18, 36, 14, 44, 12, 24, 55, 20, 50, 7, 7, 65, 18, 36, 34, 69, 46
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Multiplicative suborder of 2 (mod 2n+1) (or sord(2, 2n+1)).
For the complexity of computing this, see A002326.
It appears that under iteration of the base-n Kaprekar map, for even n > 2 (A165012, A165051, A165090, A151949 in bases 4, 6, 8, 10), almost all cycles are of length a(n/2 - 1); proved under the additional constraint that the cycle contains at least one element satisfying "number of digits (n-1) - number of digits 0 = o(total number of digits)". [From Joseph Myers (jsm(AT)polyomino.org.uk), Sep 05 2009]
From Gary W. Adamson, Sep 20 2011 (Start): a(n) can be determined by the cycle lengths of iterates using x^2 - 2, seed 2*Cos 2Pi/N; as shown in the A065941 comment of Sep 06 2011. The iterative map of the logistic equation 4x*(1-x) is likewise chaotic with the same cycle lengths but initiating the trajectory with Sin^2 2*Pi/N, N = 2n+1. [Kappraff & Adamson, 2004]. Chaotic terms with the identical cycle lengths can be obtain by applying Newton's method to i = sqrt(-1) [Strang, also Kappraff and Adamson, 2003], resulting in the morphism for the Cot 2Pi/N trajectory: (x^2-1)/2x. (end)
Roots of signed n-th row A054142 polynomials are chaotic with respect to the operation (-2, x^2), with cycle lengths A003558(n). Example: starting with a root to x^3 - 5x^2 + 6x - 1 = 0; (2 + 2*Cos 2Pi/N = 3.24697...); we obtain the trajectory (3.24697...-> 1.55495...-> .198062...); the roots to the polynomial with cycle length 3 matching A003558(3) = 3. - Gary W. Adamson, Sep 21 2011
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REFERENCES
| Jay Kappraff and Gary W. Adamson, The Relationship of the Cotangent Function to Special Relativit Theory, Silver Means, p-cycles, and Chaos Theory; FORMA, Vol. 18, No. 3, pp 249-262 (2003)
Gilbert Strang, A Chaotic Search for i, College Mathematics Journal 22, 3-12, (1991)
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..1000
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol 2 (2004), p 65.
H. J. Smith, XICalc - Extra Precision Integer Calculator.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics, Multiplicative Order.
S. Wolfram, Algebraic Properties of Cellular Automata (1984), Appendix B.
Eric Weisstein's World of Mathematics, Math World: Suborder Function
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FORMULA
| a(n) = log_2(A160657(n) + 2) - 1. - Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), May 22 2009
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EXAMPLE
| a(3) = 3 since f(x), x^2 - 2 has a 3 period using seed 2*Cos 2Pi/7, where 7 = 2*3 + 1.
a(15) = 5 since the iterative map of the logistic equation 4x*(1-x) has a period 5 using seed Sin^2 2Pi/N; N = 31 = 2*15 + 1.
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MATHEMATICA
| Suborder[a_, n_] := If[n>1 && GCD[a, n]==1, Min[MultiplicativeOrder[a, n, {-1, 1}]], 0]; Table[Suborder[2, 2n+1], {n, 0, 100}] - T. D. Noe (noe(AT)sspectra.com), Aug 02 2006
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CROSSREFS
| Cf. A054142, A065941, A085478, A160657.
Sequence in context: A023160 A085312 A046530 * A141419 A072451 A023156
Adjacent sequences: A003555 A003556 A003557 * A003559 A003560 A003561
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Harry J. Smith (hjsmithh(AT)sbcglobal.net), Feb 11 2005
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 02 2006
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