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Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function $$N_f\left(x\right)=\#\left\{0\le n\<x\left|f\right(n)\mid n\right\}$$ under a mild restriction on the values of $f$. When $f=s_q$, the base $q$ sum of digits function, the integers counted by $N_f$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.