On the counts of p-rough numbers
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913232740941824 |
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| author | Holt, Fred B. |
| author_facet | Holt, Fred B. |
| contents | The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $Φ(x,p)$, and we introduce the function $ΔΦ(x,p)$ which is the difference between $Φ(x,p)$ and the line of symmetry. $ΔΦ(x,p)$ is periodic and bounded and has a rotational symmetry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_07570 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the counts of p-rough numbers Holt, Fred B. Number Theory 11A41 (Primary), 11A51 (Secondary) The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $Φ(x,p)$, and we introduce the function $ΔΦ(x,p)$ which is the difference between $Φ(x,p)$ and the line of symmetry. $ΔΦ(x,p)$ is periodic and bounded and has a rotational symmetry. |
| title | On the counts of p-rough numbers |
| topic | Number Theory 11A41 (Primary), 11A51 (Secondary) |
| url | https://arxiv.org/abs/2308.07570 |