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A336783 Integers b where the number of cycles under iteration of sum of cubes of digits in base b is exactly four.

%I A336783 #21 Jan 09 2021 02:09:07
%S A336783 5,90,188
%N A336783 Integers b where the number of cycles under iteration of sum of cubes of digits in base b is exactly four.
%C A336783 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 A336783 Consider the map S_{x^3,b}: N to N, with S_{x^3,b}(m) := x_0^3+ ... + x_d^3.
%C A336783 It is known that the orbit set {m,S_{x^3,b}(m), S_{x^3,b}(S_{x^3,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^3,i) the set of bases b such that the set of cycles associated to S_{x^3,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^3,4).
%C A336783 Meanwhile, the known terms of the sequence L(x^3,1) is {2}, L(x^3,2) is empty, and L(x^3,3) is {3, 26}. It's undetermined whether the complete sequences are finite, if so, whether the above give all terms.
%H A336783 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 A336783 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 A336783 Integers b where A194025(b) + A194281(b) = 4.
%e A336783 For instance, when b=5 the associated four cycles are (1),(28),(118) and (9,65,35).
%Y A336783 Cf. A194025, A194281.
%Y A336783 Cf. A336744 and A336762 (sum of squares of digits).
%K A336783 nonn,base,hard,more
%O A336783 1,1
%A A336783 _Haiyang Wang_, Aug 04 2020