Partitions with prescribed sum of reciprocals: asymptotic bounds
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908463105310720 |
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| author | van Doorn, Wouter |
| author_facet | van Doorn, Wouter |
| contents | In $1963$ Graham proved that every positive integer $n \ge 78$ can be written as a sum of distinct positive integers $a_1, a_2, \ldots, a_r$ for which $\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_r}$ is equal to $1$. In the same paper he managed to further generalize this, and showed that for all positive rationals $α$ and all positive integers $m$, there exists an $n_{α, m}$ such that every positive integer $n \ge n_{α, m}$ has a partition with distinct parts, all larger than or equal to $m$, and such that the sum of reciprocals is equal to $α$. No attempt was made to estimate the quantity $n_{α, m}$, however. With $n_α := n_{α, 1}$, in this paper we provide near-optimal upper bounds on $n_α$ and $n_{α, m}$, as well as bounds on the cardinality of the set $\{α: n_α \le n\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partitions with prescribed sum of reciprocals: asymptotic bounds van Doorn, Wouter Number Theory In $1963$ Graham proved that every positive integer $n \ge 78$ can be written as a sum of distinct positive integers $a_1, a_2, \ldots, a_r$ for which $\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_r}$ is equal to $1$. In the same paper he managed to further generalize this, and showed that for all positive rationals $α$ and all positive integers $m$, there exists an $n_{α, m}$ such that every positive integer $n \ge n_{α, m}$ has a partition with distinct parts, all larger than or equal to $m$, and such that the sum of reciprocals is equal to $α$. No attempt was made to estimate the quantity $n_{α, m}$, however. With $n_α := n_{α, 1}$, in this paper we provide near-optimal upper bounds on $n_α$ and $n_{α, m}$, as well as bounds on the cardinality of the set $\{α: n_α \le n\}$. |
| title | Partitions with prescribed sum of reciprocals: asymptotic bounds |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.02200 |