Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3 x + 1 Problem)

Given p ∈ N = {1, 2, 3 ...}, as a generalization of the 3 x + 1 problem [7], we study the behavior of the sequences s ( m ) = { m n } n ≥ 0 , m ∈ Z (the set of the integers), defined by the iterative formula and a r = t r /p ( t r ∈ N), and b r are chosen in such a way that m n ∈ Z for every n . Our aim is to establish when these sequences are divergent, or convergent into a cycle, considering also a fixed point as a cycle. Moreover, the structure of possible cycles is investigated.