A Class of Bounded Iterative Sequences of Integers.

In this note, we show that, for any real number τ ∈ [ 1 2 , 1) , any finite set of positive integers K and any integer s 1 ≥ 2 , the sequence of integers s 1 , s 2 , s 3 , ... satisfying s i + 1 − s i ∈ K if s i is a prime number, and 2 ≤ s i + 1 ≤ τ s i if s i is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on τ , s 1 and the maximal element of K only. In particular, if K is a singleton set and for each composite s i the integer s i + 1 in the interval [ 2 , τ s i ] is chosen by some prescribed rule, e.g., s i + 1 is the largest prime divisor of s i , then the sequence s 1 , s 2 , s 3 , ... is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either K = { 1 } and τ ∈ [ 1 2 , 3 4) or K = { 2 } and τ ∈ [ 1 2 , 5 9) . [ABSTRACT FROM AUTHOR]