This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336744 #42 Jan 08 2021 22:16:58
%S A336744 14,66,94,300,384,436,496,750,1406,1794,2336,2624,28034
%N A336744 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly three.
%C A336744 Let b > 1 be an integer, and write the base b expansion of any nonnegative integer m as m = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.
%C A336744 Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(m) := x_0^2+ ... + x_d^2.
%C A336744 This is the 'sum of the squares of the digits' dynamical system alluded to in the name of the sequence.
%C A336744 It is known that the orbit set {m,S_{x^2,b}(m), S_{x^2,b}(S_{x^2,b}(m)), ...} is finite for all m>0. Each orbit contains a finite cycle, and for a given base b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^2,i) the set of bases b such that the set of cycles associated to S_{x^2,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^2,3).
%C A336744 A 1978 conjecture of Hasse and Prichett describes the set L(x^2,2). New elements have been added to this set in the paper Integer Dynamics, by D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang. It is natural to wonder whether the set L(x^2,3) is infinite. It is a folklore conjecture that L(x^2,1) = {2,4}.
%H A336744 H. Hasse and G. Prichett, <a href="https://doi.org/10.1515/crll.1978.298.8">A conjecture on digital cycles</a>, J. reine angew. Math. 298 (1978), 8--15. Also on <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002194481">GDZ</a>.
%H A336744 D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, <a href="http://alpha.math.uga.edu/~lorenz/IntegerDynamics.pdf">Integer Dynamics</a>, preprint.
%F A336744 Integers b such that A193583(b)+A193585(b) = 3. - _Michel Marcus_, Aug 03 2020
%e A336744 For instance, in base 14, the three cycles are (1), (37,85), and (25,122,164,221,123,185,178,244,46). To verify that (37,85) is a cycle in base 14, note that 37=9+2*14, and that 9^2+2^2=85. Similarly, 85=1+6*14, and 1^2+6^2=37.
%Y A336744 Cf. A193583, A193585 (where cycles and fixed points are treated separately).
%Y A336744 Cf. A336762 (2 cycles).
%Y A336744 Cf. A336783 (4 cycles with sum of cubes of the digits).
%K A336744 nonn,base,hard,more
%O A336744 1,1
%A A336744 _Dino Lorenzini_, Aug 02 2020