In this note, we study the arithmetic function
![$ f : \mathbb{Z}_+^*
\rightarrow \mathbb{Q}_+^*$](img2.gif)
defined by
![$ f(2^k \ell) = \ell^{1 -
k}$](img3.gif)
(
![$ \forall k , \ell \in \mathbb{N}$](img4.gif)
,
![$ \ell$](img5.gif)
odd). We show
several important properties about this function, and we use
them to obtain some curious results involving the
![$ 2$](img6.gif)
-adic
valuation. In the last section of the paper, we generalize those results to any other
![$ p$](img7.gif)
-adic valuation.