General formulas of global characteristic coefficients of Collatz function

TL;DR

This paper derives three general formulas for global characteristic coefficients of the Collatz function, providing a mathematical framework to analyze its iterative behavior on natural numbers.

Contribution

It introduces three global characteristic coefficients for the Collatz function and derives their general expressions as functions of the iteration count n.

Findings

Formulas for three global characteristic coefficients of Collatz function

Expressions of these coefficients as functions of iteration number n

Mathematical characterization of Collatz function iterations

Abstract

The purpose of this paper is to show three general formulas of three global characteristic coefficients of Collatz function. The Collatz function is defined by the following operation on an arbitrary positive integer if N is odd multiply it by 3 and add 1 then the sum obtained is divided by 2, if N is even divide it by 2. Based on the principle, we define the n-order function denoted by T^n such as the different expressions of that function are results of applying Collatz function n times to a natural numbers which are expressed in well determined forms. Based on these expressions, we can characterize that n-order function by three global characteristic coefficients. In the first, we define these three global coefficients. Secondly, we show that each global characteristic coefficient has a general expression as a function of n.