New results on the stopping time behaviour of the Collatz 3x + 1 function

TL;DR

This paper investigates the distribution of starting numbers with finite stopping times in the Collatz 3x+1 problem, providing a conjectured formula for counting residue classes and an iterative method to compute these counts.

Contribution

It introduces a new conjectured formula for the number of residue classes with finite stopping times and an iterative algorithm to compute these counts for the Collatz problem.

Findings

Derived a conjectured formula for z_n based on factorial and binomial coefficients.

Developed an iterative algorithm to generate z_n values for n > 6.

Connected the residue class counts to the OEIS sequence A100982.

Abstract

Let $\sigma_n=\lfloor1+n\cdot\log_23\rfloor$. For the Collatz 3x + 1 function exists for each $n\in\mathbb{N}$ a set of different residue classes $(\text{mod}\ 2^{\sigma_n})$ of starting numbers $s$ with finite stopping time $\sigma(s)=\sigma_n$. Let $z_n$ be the number of these residue classes for each $n\geq0$ as listed in the OEIS as A100982. It is conjectured that for each $n\geq4$ the value of $z_n$ is given by the formula \begin{align*} z_n=\frac{(m+n-2)!}{m!\cdot(n-2)!}-\sum_{i=2}^{n-1}\binom{\big\lfloor\frac{3(n-i)+\delta}{2}\big\rfloor}{n-i}\cdot z_i, \end{align*} where $m=\big\lfloor(n-1)\cdot\log_23\big\rfloor-(n-1)$ and $\delta\in\mathbb{Z}$ assumes different values within the sum at intervals of 5 or 6 terms. This allows us to create an iterative algorithm which generates $z_n$ for each $n>6$.