OFFSET
1,1
COMMENTS
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.
Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(m) := x_0^2+ ... + x_d^2.
This is the 'sum of the squares of the digits' dynamical system alluded to in the name of the sequence.
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).
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}.
LINKS
H. Hasse and G. Prichett, A conjecture on digital cycles, J. reine angew. Math. 298 (1978), 8--15. Also on GDZ.
D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, Integer Dynamics, preprint.
FORMULA
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,base,hard,more
AUTHOR
Dino Lorenzini, Aug 02 2020
STATUS
approved