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 A336762 #23 Jan 08 2021 22:17:01
%S A336762 6,10,16,20,26,40,8626,481360
%N A336762 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly two.
%C A336762 Let b > 1 be an integer, and write the base b expansion of any nonnegative integer n as n = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.
%C A336762 Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(n) := x_0^2+ ... + x_d^2.
%C A336762 It is known that the orbit set {n, S_{x^2,b}(n), S_{x^2,b}(S_{x^2,b}(n)), ...} is finite for all n > 0. Each orbit contains a finite cycle, and for a given 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^2,2).
%C A336762 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. The sequence contains all b <= 10^6 that are in L(x^2,2).
%H A336762 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 A336762 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.
%e A336762 For instance, b = 10 is in this sequence since in the decimal system, there are exactly two cycles (1) and (4, 16, 37, 58, 89, 145, 42, 20).
%Y A336762 Cf. A193583 and A193585 (b is in this sequence if A193583(b)+A193585(b) = 2).
%Y A336762 Cf. A336744 (3 cycles).
%Y A336762 Cf. A336783 (4 cycles with sum of cubes of the digits).
%K A336762 nonn,base,hard,more
%O A336762 1,1
%A A336762 _Makoto Suwama_, Aug 03 2020