Partitions with prescribed sum of reciprocals: computational results
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909702930038784 |
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| author | van Doorn, Wouter |
| author_facet | van Doorn, Wouter |
| contents | For a positive rational $α$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $α$-partition of $n$, if the sum of the $a_i$ is equal to $n$ and the sum of the reciprocals of the $a_i$ is equal to $α$. Define $n_α$ to be the smallest positive integer such that for all $n \ge n_α$ an $α$-partition of $n$ exists and, for a positive integer $M \ge 2$, define $N_M$ to be the smallest positive integer such that for all $n \ge N_M$ a $1$-partition of $n$ exists where $M$ does not divide any of the $a_i$. In this paper we determine $N_M$ for all $M \ge 2$, and find the set of all $α$ such that $n_α \le 100$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partitions with prescribed sum of reciprocals: computational results van Doorn, Wouter Number Theory For a positive rational $α$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $α$-partition of $n$, if the sum of the $a_i$ is equal to $n$ and the sum of the reciprocals of the $a_i$ is equal to $α$. Define $n_α$ to be the smallest positive integer such that for all $n \ge n_α$ an $α$-partition of $n$ exists and, for a positive integer $M \ge 2$, define $N_M$ to be the smallest positive integer such that for all $n \ge N_M$ a $1$-partition of $n$ exists where $M$ does not divide any of the $a_i$. In this paper we determine $N_M$ for all $M \ge 2$, and find the set of all $α$ such that $n_α \le 100$. |
| title | Partitions with prescribed sum of reciprocals: computational results |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.01409 |