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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.04342 |
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| _version_ | 1866913257099362304 |
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| author | Lagarias, Jeffrey C. Richman, David Harry |
| author_facet | Lagarias, Jeffrey C. Richman, David Harry |
| contents | An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the $a$-floor quotient relations $\,\preccurlyeq_{a}\,$, for $a \in \mathbb{N}^+$, which interpolate between the floor quotient order and the divisor order on $\mathbb{N}^+$. The paper studies the internal structure of these orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04342 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The family of $a$-floor quotient partial orders Lagarias, Jeffrey C. Richman, David Harry Number Theory 06A06, 11A05 (Primary) 05A16, 15B36, 26D07 (Secondary) An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the $a$-floor quotient relations $\,\preccurlyeq_{a}\,$, for $a \in \mathbb{N}^+$, which interpolate between the floor quotient order and the divisor order on $\mathbb{N}^+$. The paper studies the internal structure of these orders. |
| title | The family of $a$-floor quotient partial orders |
| topic | Number Theory 06A06, 11A05 (Primary) 05A16, 15B36, 26D07 (Secondary) |
| url | https://arxiv.org/abs/2403.04342 |