Runs of Consecutive Integers Having the Same Number of Divisors
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916350633443328 |
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| author | Spătaru, Vlad-Titus |
| author_facet | Spătaru, Vlad-Titus |
| contents | Our objective is to provide an upper bound for the length $\ell_N$ of the longest run of consecutive integers smaller than $N$ which have the same number of divisors. We prove in an elementary way that $\log\ell_N\ll(\log N\log\log N)^λ$, where $λ=1/2$. Using estimates for the Jacobsthal function, we then improve the result to $λ=1/3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_04464 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Runs of Consecutive Integers Having the Same Number of Divisors Spătaru, Vlad-Titus Number Theory 11A25, 11N37 (Primary) Our objective is to provide an upper bound for the length $\ell_N$ of the longest run of consecutive integers smaller than $N$ which have the same number of divisors. We prove in an elementary way that $\log\ell_N\ll(\log N\log\log N)^λ$, where $λ=1/2$. Using estimates for the Jacobsthal function, we then improve the result to $λ=1/3$. |
| title | Runs of Consecutive Integers Having the Same Number of Divisors |
| topic | Number Theory 11A25, 11N37 (Primary) |
| url | https://arxiv.org/abs/2301.04464 |