Irregular Stanley sequences plausibly do not have growth $\Theta(n^2/\log n)$

Abstract

Stanley sequences starting from the set $\{0, n\}$ where $n$ is a positive integer have long been conjectured to be divided into two types: the "regular" type where the growth rate is $\Theta(n^{\log_2(3)})$, and the "irregular" type where the growth rate is thought to be $\Theta(n^2/\log n)$. A paradigmatic case of a candidate irregular type is $n=4$, although to date no value of $n$ has been proven to have such a growth rate. Here, we provide strong numerical evidence against this conjectured growth rate for $n=4$. Specifically, for $n=4$, it seems plausible that the upper bound is $O(n^2/\log n)$ but that the lower bound is in fact $\Omega(n^{2-\delta})$ for some $\delta > 0$. This appears to be because the sequence is not totally "random" as has been assumed. Limitations of the numerical method here is discussed.