A Study of a Curious Arithmetic Function

In this note, we study the arithmetic function f : Z*(+) -> Q*(+) defined by f(2(k) l)=l(1-k) (for all k, l is an element of N , l odd). We show several important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation. In the last section of the paper, we generalize those results to any other rho-adic valuation.