On the periodicity of some Farhi arithmetical functions
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- by Qing-Zhong Ji and Chun-Gang Ji PDF
- Proc. Amer. Math. Soc. 138 (2010), 3025-3035 Request permission
Abstract:
Let $k\in \mathbb {N}$. Let $f(x)\in \mathbb {Z}[x]$ be any polynomial such that $f(x)$ and $f(x+1)f(x+2)\cdots f(x+k)$ are coprime in $\mathbb {Q}[x]$. We call \[ g_{k,f}(n):=\frac {|f(n)f(n+1)\cdots f(n+k)|} {\text {lcm}(f(n),f(n+1),\cdots ,f(n+k))}\] a Farhi arithmetic function. In this paper, we prove that $g_{k,f}$ is periodic. This generalizes the previous results of Farhi and Kane, and Hong and Yang.References
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Additional Information
- Qing-Zhong Ji
- Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- Email: qingzhji@nju.edu.cn
- Chun-Gang Ji
- Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
- Email: cgji@njnu.edu.cn
- Received by editor(s): June 8, 2009
- Published electronically: April 27, 2010
- Additional Notes: The first author was partially supported by Grants No. 10571080 and 10871088 from the NNSF of China.
The second author was partially supported by Grants No. 10971098 and 10771103 from the NNSF of China. - Communicated by: Wen-Ching Winnie Li
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 3025-3035
- MSC (2010): Primary 11A25; Secondary 11B83
- DOI: https://doi.org/10.1090/S0002-9939-10-10408-0
- MathSciNet review: 2653927