On a new additive arithmetic function related to a fixed integer

The main purpose of this paper is to define a new additive arithmetic function related to a fixed integer k\geq 1 and to study some of its properties. This function is given by \begin{equation*} \ f_{k}\left( 1\right) =0\text{ and }f_{k}\left( n\right) =\sum_{p^{\alpha}\parallel n}\left( k,\alpha \right) , \end{equation*} such that (a, b) denotes the greatest common divisor of the integers a and b.