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.

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

Table of n, a(n) for n=1..36.

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

Adjacent sequences: A329466 A329467 A329468 * A329470 A329471 A329472

Keyword

nonn,fini,full

Author

Amiram Eldar, Nov 13 2019

Extensions

Name corrected by Rémy Sigrist, Nov 14 2019

Status

approved