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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the periodicity of some Farhi arithmetical functions


Authors: Qing-Zhong Ji and Chun-Gang Ji
Journal: Proc. Amer. Math. Soc. 138 (2010), 3025-3035
MSC (2010): Primary 11A25; Secondary 11B83
Published electronically: April 27, 2010
MathSciNet review: 2653927
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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

$\displaystyle g_{k,f}(n):=\frac{\vert f(n)f(n+1)\cdots f(n+k)\vert} {\text{lcm}(f(n),f(n+1),\cdots,f(n+k))}$

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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10408-0
PII: S 0002-9939(10)10408-0
Keywords: Arithmetical functions, least common multiple, periodicity
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.