On the periodicity of some Farhi arithmetical functions

• Published in: arXiv.org
• Main Authors: Qing-Zhong Ji, Chun-Gang, Ji
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.05.2009
• Subjects: Mathematical analysis; Mathematical functions; Periodic variations; Polynomials

Let \(k\in\mathbb{N}\). Let \(f(x)\in \Bbb{Z}[x]\) be any polynomial such that \(f(x)\) and \(f(x+1)f(x+2)... f(x+k)\) are coprime in \(\mathbb{Q}[x]\). We call $$g_{k,f}(n):=\frac{|f(n)f(n+1)... f(n+k)|}{\text{lcm}(f(n),f(n+1),...,f(n+k))}$$ a Farhi arithmetic function. In this paper, we prove that...