A Family of Iterated Maps on Natural Numbers
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910765024280576 |
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| author | Das, Angsuman |
| author_facet | Das, Angsuman |
| contents | In this paper, we introduce and study the iterates of the following family of functions $φ_k$ defined on natural numbers which exhibits nice properties. $$φ_k(x)=\left\lbrace \begin{array}{ll} x+k, & \mbox{ if $x$ is prime;}\\ \mbox{largest prime divisor of $x$,} & \mbox{ if $x$ is composite;} \end{array} \right.$$ In particular, we study the periodic behaviour of the trajectories of these iterated functions. In some cases, we provide proofs of these properties and in some other cases we pose some open problems based on numerical evidences supported by heuristic arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06629 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Family of Iterated Maps on Natural Numbers Das, Angsuman Number Theory 11B83, 11B37, 11A25 In this paper, we introduce and study the iterates of the following family of functions $φ_k$ defined on natural numbers which exhibits nice properties. $$φ_k(x)=\left\lbrace \begin{array}{ll} x+k, & \mbox{ if $x$ is prime;}\\ \mbox{largest prime divisor of $x$,} & \mbox{ if $x$ is composite;} \end{array} \right.$$ In particular, we study the periodic behaviour of the trajectories of these iterated functions. In some cases, we provide proofs of these properties and in some other cases we pose some open problems based on numerical evidences supported by heuristic arguments. |
| title | A Family of Iterated Maps on Natural Numbers |
| topic | Number Theory 11B83, 11B37, 11A25 |
| url | https://arxiv.org/abs/2312.06629 |