MATHEMATICA BOHEMICA, Vol. 142, No. 1, pp. 21-25, 2017

On a certain class of arithmetic functions

Antonio M. Oller-Marcén

Received October 20, 2014.   First published October 18, 2016.

Antonio Miguel Oller-Marcén, Centro Universitario de la Defensa de Zaragoza, Academia General Militar. Ctra. de Huesca s/n., C. P. 50090 Zaragoza, Spain, e-mail:

Abstract: A homothetic arithmetic function of ratio $K$ is a function $f \mathbb{N}\rightarrow R$ such that $f(Kn)=f(n)$ for every $n\in\mathbb{N}$. Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f(\mathbb{N})$ in terms of the period and the ratio of $f$.

Keywords: arithmetic function; periodic function; homothetic function

Classification (MSC 2010): 11A25, 11B99

DOI: 10.21136/MB.2017.0071-14

Full text available as PDF.

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