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A329469
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Perfectly cyclic numbers: numbers k such that the iterations of the mapping x -> f(x) = x^2 + c (mod k), starting at x = f(c), is purely periodic for all 0 <= c <= k.
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0
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1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 25, 30, 36, 40, 45, 50, 60, 72, 75, 90, 100, 120, 150, 180, 200, 225, 300, 360, 450, 600, 900, 1800
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OFFSET
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1,2
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COMMENTS
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Fletcher and Smith proved that there are 36 terms in this sequence.
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LINKS
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Graham Everest, Alfred J. Van Der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, Mathematical Surveys and Monographs, Vol. 104, Providence, RI: American Mathematical Society, 2003, pp. 59-61, draft.
Matthew Fletcher and Geoff C. Smith, Chaos, elliptic curves and all that, in: G. E. Bergum, A. N. Philippou, and A. F. Horadam (eds.), Applications of Fibonacci numbers, Vol. 5, Proceedings of The Fifth International Conference on Fibonacci Numbers and Their Applications, The University of St. Andrews, Scotland, July 20-July 24, 1992, Springer, Dordrecht, 1993, pp. 245-256.
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FORMULA
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Numbers of the form 2^a * 3^b * 5^c, where 0 <= a <= 3, 0 <= b, c <= 2.
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EXAMPLE
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3 is in the sequence since {f(c), f(f(c)), ....} = {0, 0, 0, ... } for c = 0 and 3, {2, 2, 2, ... } for c = 1, and {0, 2, 0, 2, ... } for c = 2, are all purely periodic.
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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