More on Counting Sequences

This note can be viewed as a logical continuation of our previous Note published earlier in the Monthly 1]. We shall repeat here, brieey, its main results to make the reading easier. A counting sequence S is a sequence of sequences fS i g 1 i=0 of positive integers. The sequence S i+1 is obtained from S i by counting the number m k of times an integer k occurs in S i and writing down in S i+1 the pairs m k ; k in increasing order of the factors k, for all k for which m k > 0. In 1] the Ultimate Periodicity Theorem was proved: Ultimate Periodicity Theorem. Beginning with any nite initial sequence S 0 , all the sequences S i 2 S have bounded length, and S is ultimately periodic. This fact means that every counting sequence eventually \stabilises". The question we are going to study in this note is what are those \stable" points? We shall use the term recurring to denote them, so a sequence is called a recurring point if it occurs among the successors of itself. What are the properties that allow these points to be stable? Can we get a complete list of them? To answer these questions we recall the notation from the previous Note, mentioning brieey the basic facts we have established there. where the elements m j (i) (1 j n) are called multipliers and the elements f j (i) | factors.