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

Ji, Qingzhong; Ji, Chungang

Proc. Am. Math. Soc. 138, No. 9, 3025-3035 (2010).

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.

Reviewer: Olaf Ninnemann (Berlin)

MSC:

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

Keywords:

arithmetical functions; least common multiple of consecutive integers; periodicity

Citations:

Zbl 1229.11007; Zbl 1213.11014

References:

[1]	Bakir Farhi, Minorations non triviales du plus petit commun multiple de certaines suites finies d’entiers, C. R. Math. Acad. Sci. Paris 341 (2005), no. 8, 469 – 474 (French, with English and French summaries). · Zbl 1117.11005
[2]	Bakir Farhi, Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory 125 (2007), no. 2, 393 – 411. · Zbl 1124.11005
[3]	Bakir Farhi and Daniel Kane, New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc. 137 (2009), no. 6, 1933 – 1939. · Zbl 1229.11007
[4]	Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481 – 547. · Zbl 1191.11025
[5]	G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. · Zbl 0423.10001
[6]	Shaofang Hong and Yujuan Yang, On the periodicity of an arithmetical function, C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 717 – 721 (English, with English and French summaries). · Zbl 1213.11014