A $2$-Regular Sequence That Counts The Divisors of $n^2 + 1$
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911234145648640 |
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| author | Shakov, Anton |
| author_facet | Shakov, Anton |
| contents | We introduce the $2$-regular integer sequence A383066 $= (s(n))_{n \geq 1}$, which begins $0, 1, 1, 2, 3, 3, 2, \ldots$. We prove that the number of occurrences of an integer $m \geq 0$ in this sequence is equal to $τ(m^2+1)$, the number of divisors of $m^2 + 1$. Using this fact, we give a generating function for $τ(m^2+1)$. We also discuss other interesting properties of $s(n)$, including its relationship to the Fibonacci sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A $2$-Regular Sequence That Counts The Divisors of $n^2 + 1$ Shakov, Anton Number Theory Primary 11B37, Secondary 11A25, 05C05, 11B39 We introduce the $2$-regular integer sequence A383066 $= (s(n))_{n \geq 1}$, which begins $0, 1, 1, 2, 3, 3, 2, \ldots$. We prove that the number of occurrences of an integer $m \geq 0$ in this sequence is equal to $τ(m^2+1)$, the number of divisors of $m^2 + 1$. Using this fact, we give a generating function for $τ(m^2+1)$. We also discuss other interesting properties of $s(n)$, including its relationship to the Fibonacci sequence. |
| title | A $2$-Regular Sequence That Counts The Divisors of $n^2 + 1$ |
| topic | Number Theory Primary 11B37, Secondary 11A25, 05C05, 11B39 |
| url | https://arxiv.org/abs/2510.22805 |