A Study of a Curious Arithmetic Function

The purpose of this paper is to study the arithmetic function f : Z+ → Q ∗ + defined by f(2l) = l (∀k, l ∈ N, l odd). We have, for example, f(1) = 1, f(2) = 1, f(3) = 3, f(12) = 1 3 , f(40) = 1 25 , . . . , so it is clear that f(n) is not always an integer. However, we will show in what follows that f satisfies the property that the product of the f(r) for 1 ≤ r ≤ n is always an integer, and it is a multiple of all odd prime numbers not exceeding n. Further, we exploit the properties of f to establish some curious properties concerning the 2-adic valuation. In the last section of the paper, we give (without proof) the analogous properties for other p-adic valuations. The study of f requires introducing the two auxiliary arithmetic functions g : Q+ → Z ∗ + and h : Z+ → Q ∗ +, defined by: