A study of a curious arithmetic function

TL;DR

This paper investigates a unique arithmetic function defined on positive integers, exploring its properties and deriving interesting results related to the 2-adic valuation.

Contribution

It introduces and analyzes a novel arithmetic function, revealing new properties and applications involving 2-adic valuation.

Findings

The function exhibits specific algebraic properties.

Connections to 2-adic valuation are established.

Curious results are derived from the function's properties.

Abstract

In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation.