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A Study of a Curious Arithmetic Function Bakir Farhi
Department of Mathematics
University of Béjaia
Béjaia
Algeria
mailto:bakir.farhi@gmail.combakir.farhi@gmail.com

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Abstract:

In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \rightarrow \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ( $\forall k , \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about this function, and 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 $p$-adic valuation.