On the congruence properties and growth rate of a recursively defined sequence
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914833874550784 |
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| author | van Doorn, Wouter |
| author_facet | van Doorn, Wouter |
| contents | Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7 \}$, then the natural density of $n$ such that $a_n \equiv x \pmod{8}$ exists and equals $\frac{1}{6}$. Next we will prove that if $m \le 15$ is not divisible by $4$, then the lower density of $n$ such that $a_n$ is divisible by $m$, is strictly positive. To put these results in a broader context, we will then posit a general conjecture about the density of $n$ such that $a_n \equiv x \pmod{m}$ for any given $x$ and any $m$ not divisible by $32$. Finally, we will show that there exists a function $f$ such that $n^{f(n)} < a_n < n^{f(n) + ε}$ for all $ε> 0$ and all large enough $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the congruence properties and growth rate of a recursively defined sequence van Doorn, Wouter Number Theory Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7 \}$, then the natural density of $n$ such that $a_n \equiv x \pmod{8}$ exists and equals $\frac{1}{6}$. Next we will prove that if $m \le 15$ is not divisible by $4$, then the lower density of $n$ such that $a_n$ is divisible by $m$, is strictly positive. To put these results in a broader context, we will then posit a general conjecture about the density of $n$ such that $a_n \equiv x \pmod{m}$ for any given $x$ and any $m$ not divisible by $32$. Finally, we will show that there exists a function $f$ such that $n^{f(n)} < a_n < n^{f(n) + ε}$ for all $ε> 0$ and all large enough $n$. |
| title | On the congruence properties and growth rate of a recursively defined sequence |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.09532 |