Classification of Collatz infinite sequences

TL;DR

This paper classifies Collatz infinite sequences based on the asymptotic behavior of their defining coefficients, providing insights into their long-term behavior without proving the Collatz conjecture.

Contribution

It introduces a novel classification scheme for Collatz sequences based on the limits of their coefficient functions, enhancing understanding of their infinite behavior.

Findings

Classification of sequences based on coefficient limits

Determination of proportions of each sequence class

Insights into Collatz sequence behavior as n approaches infinity

Abstract

In the present paper, we are interested in classifying of Collatz sequences on based to the different behavior of these sequences when their lengths tend to infinity. A Collatz infinite sequence can be defined as an infinite ordered set of positive integers such that the term of rank n is results of applying Collatz map n times to the first term. Such term can be expressed on the form Tn(P)=A(P,n)P+B(P,n). When n tends to infinity, each function among the two partial coefficients denoted by A(P,n) and B(P,n) behaves in different ways. This allows us to determine all categories of Collatz infinite sequences. First, we carry out a classification of Collatz infinite sequences on based of the different possible limits of the two coefficients. In second time, we determine the different proportions of every class of the infinite sequences. Note that results obtained do not represent a proof of a Collatz conjecture but they have a strong relationship with this conjecture and it allows us to better understand the behavior of Collatz sequences when n tend to infinity.