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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.16072 |
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| _version_ | 1866929423109849088 |
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| author | Heyman, Randell Miraj, MD Rahil |
| author_facet | Heyman, Randell Miraj, MD Rahil |
| contents | Let $X$ be a positive integer and $t$ a real number great than 1. The family of sets $\left\{\big\lfloor\frac{X}{n^t}\big\rfloor ~:~ 1\leq n\leq X\right\}$ have an interesting prime distribution property. We give an exact formula for the cardinality of these sets. We provide an estimate for the cardinality of the set $\left\{\big\lfloor\frac{X}{p}\big\rfloor ~:~ p~ \text{prime},~ p\leq X\right\}$. For positive real $X$, we derive asymptotic formulas for the cardinality of the set $\big\{\lfloor f(n)\rfloor ~:~ 1\leq n\leq X\big\}$ for various sets of functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16072 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On some floor function sets Heyman, Randell Miraj, MD Rahil Number Theory Let $X$ be a positive integer and $t$ a real number great than 1. The family of sets $\left\{\big\lfloor\frac{X}{n^t}\big\rfloor ~:~ 1\leq n\leq X\right\}$ have an interesting prime distribution property. We give an exact formula for the cardinality of these sets. We provide an estimate for the cardinality of the set $\left\{\big\lfloor\frac{X}{p}\big\rfloor ~:~ p~ \text{prime},~ p\leq X\right\}$. For positive real $X$, we derive asymptotic formulas for the cardinality of the set $\big\{\lfloor f(n)\rfloor ~:~ 1\leq n\leq X\big\}$ for various sets of functions. |
| title | On some floor function sets |
| topic | Number Theory |
| url | https://arxiv.org/abs/2309.16072 |