cp's OEIS Frontend

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.

Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(n) := x_0^2+ ... + x_d^2.

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).

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).

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^3,b}: N to N, with S_{x^3,b}(m) := x_0^3+ ... + x_d^3.

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).

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.