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

Abstract

Let ${\sigma }_n=\lfloor 1+n\cdot {\log }_{2}3\rfloor$. 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 $n\ge 0$ as listed in the OEIS as A100982. It is conjectured that for each $n\ge 4$ 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)+δ}{2}\big\rfloor}{n-i}\cdot z_i, \end{align*} where $m=\lfloor (n-1)\cdot {\log }_{2}3\rfloor -(n-1)$ 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 $n>6$.

arXiv comment18 pages, 1 figure, algorithms/programs in PARI/GP