On the periodicity of some Farhi arithmetical functions. (English) Zbl 1246.11013

In this paper the authors generalize results of B. Farhi and D. Kane [Proc. Am. Math. Soc. 137, No. 6, 1933–1939 (2009; Zbl 1229.11007)], and S. Hong and Y. Yang [C. R., Math., Acad. Sci. Paris 346, No. 13–14, 717–721 (2008; Zbl 1213.11014)] concerning the periodicity of the so-called Farhi arithmetic function. 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 \(\mathbb{Q}[x]\). Then \[ g_{k,f}(n):=\frac{|f(n)f(n+1)\cdots f(n+k)|}{\text{lcm}(f(n),f(n+1),\cdots,f(n+k))} \] is called a Farhi arithmetic function. Here the authors prove that \(g_{k,f}\) is periodic. Moreover they prove several results on the least period of the (Farhi) arithmetic function.


11A25 Arithmetic functions; related numbers; inversion formulas
11B83 Special sequences and polynomials
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