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A329469
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.
0
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
OFFSET
1,2
COMMENTS
Fletcher and Smith proved that there are 36 terms in this sequence.
LINKS
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.
FORMULA
Numbers of the form 2^a * 3^b * 5^c, where 0 <= a <= 3, 0 <= b, c <= 2.
EXAMPLE
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.
CROSSREFS
Subsequence of A051037.
Sequence in context: A029461 A128168 A232823 * A018412 A061945 A029509
KEYWORD
nonn,fini,full
AUTHOR
Amiram Eldar, Nov 13 2019
EXTENSIONS
Name corrected by Rémy Sigrist, Nov 14 2019
STATUS
approved