Irregular Stanley sequences plausibly do not have growth $Θ(n^2/\log n)$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917144122359808 |
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| author | Sothanaphan, Nat |
| author_facet | Sothanaphan, Nat |
| contents | 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 $Θ(n^{\log_2(3)})$, and the "irregular" type where the growth rate is thought to be $Θ(n^2/\log n)$. 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 $Ω(n^{2-δ})$ for some $δ> 0$. This appears to be because the sequence is not totally "random" as has been assumed. Limitations of the numerical method here is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11983 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irregular Stanley sequences plausibly do not have growth $Θ(n^2/\log n)$ Sothanaphan, Nat Number Theory Numerical Analysis 11B25 (Primary) 11Y55 (Secondary) 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 $Θ(n^{\log_2(3)})$, and the "irregular" type where the growth rate is thought to be $Θ(n^2/\log n)$. 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 $Ω(n^{2-δ})$ for some $δ> 0$. This appears to be because the sequence is not totally "random" as has been assumed. Limitations of the numerical method here is discussed. |
| title | Irregular Stanley sequences plausibly do not have growth $Θ(n^2/\log n)$ |
| topic | Number Theory Numerical Analysis 11B25 (Primary) 11Y55 (Secondary) |
| url | https://arxiv.org/abs/2512.11983 |