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For all natural numbers $n$, let $(B_{n},*_{n})$ be the algebraic structure with underlying set $\{1,\dots,n\}$ where

  1. $x*_{n}1=x+1\mod n$,

  2. $n*_{n}y=y$, and

  3. $x*_{n}(y+1)=(x*_{n}y)*_{n}(x+1)$ for $x<n,y<n$.

The algebra $(B_{n},*_{n})$ is called the $n$-th bad Laver table. For each $n$ and $x\in\{1,\dots,n\}$, there exists some number $\pi_{n}(x)$ such that $x*_{n}1<\dots<x*_{n}\pi_{n}(x)$ and where $x*_{n}y=x*_{n}z$ whenever $y=z\mod \pi_{n}(x)$. If $n=2^{N}$, then $(B_{n},*_{n})$ is known as the $N$-th classical Laver table.

Let $\sigma_{0}(x)$ denote the number of divisors of the number $x$. Here are a few unexplained observations about the bad Laver tables.

  1. For all $n$, the linear correlation coefficient between $\sigma_{0}(x)$ and $\log(\pi_{n}(x))$ is quite high. For example, if $n=339849$, then the linear correlation coefficient between $\sigma_{0}(x)$ and $\log(\pi_{n}(x))$ is $0.436963$. The linear correlation coefficient between $\sigma_{0}(x)$ and $\log(\pi_{n}(x))$ seems to depend mainly on $\gcd(2^{\infty},n)$.

  2. For all $n$, if $\alpha$ is a highly composite number, then the linear correlation coefficient between $\log(\pi_{n}(x))$ and $\log(\pi_{n}(x+\alpha))$ is also quite high. For example, if $\alpha=7200$ and $n=367611$, then the linear correlation coefficient between $\log(\pi_{n}(x))$ and $\log(\pi_{n}(x+\alpha))$ is $0.526511$.

  3. If $x<n$ and $\pi_{n}(x)=\max(\{\pi_{n}(y)|y<n\})$, then $\sigma_{0}(x)$ is often exceptionally large.

What are some mathematical explanations for these phenomena?

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  • $\begingroup$ Let $\ell_{n}$ denote the linear correlation coefficient between $\sigma_{0}(x)$ and $\log(\pi_{n}(x))$. Then the distribution of the values $\ell_{n}$ does not follow a normal distribution where the sample is taken from the odd numbers from n=30000 to n=36000. $\endgroup$ – Joseph Van Name Feb 10 at 23:21

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