Divisibility and Sequence Properties of $σ^+$ and $φ^+$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908702801395712 |
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| author | Mandal, Sagar |
| author_facet | Mandal, Sagar |
| contents | Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $φ(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient function $φ^{*}(n)$, we investigate analogous divisibility problems involving the functions $σ(n)$, $σ^{+}(n)$, and $φ^{+}(n)$. Further, we establish some interesting properties of the sequences $\left\{σ^+(n)\right\}_{n=1}^\infty$ and $\left\{φ^+(n)\right\}_{n=1}^\infty$, in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divisibility and Sequence Properties of $σ^+$ and $φ^+$ Mandal, Sagar General Mathematics 11A25 Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $φ(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient function $φ^{*}(n)$, we investigate analogous divisibility problems involving the functions $σ(n)$, $σ^{+}(n)$, and $φ^{+}(n)$. Further, we establish some interesting properties of the sequences $\left\{σ^+(n)\right\}_{n=1}^\infty$ and $\left\{φ^+(n)\right\}_{n=1}^\infty$, in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length $3$. |
| title | Divisibility and Sequence Properties of $σ^+$ and $φ^+$ |
| topic | General Mathematics 11A25 |
| url | https://arxiv.org/abs/2508.11660 |