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

Paper

Abstract

Stanley sequences starting from the set

where

is a positive integer have long been conjectured to be divided into two types: the "regular" type where the growth rate is

, and the "irregular" type where the growth rate is thought to be

. A paradigmatic case of a candidate irregular type is

, although to date no value of

has been proven to have such a growth rate. Here, we provide strong numerical evidence against this conjectured growth rate for

. Specifically, for

, it seems plausible that the upper bound is

but that the lower bound is in fact

for some

. This appears to be because the sequence is not totally "random" as has been assumed. Limitations of the numerical method here is discussed.