Further results on a curious arithmetic function (Q827171)

Summary: Let \(p\) be an odd prime number and \(n\) be a positive integer. Let \(v_p(n)\), \(\mathbb N^*\), and \(\mathbb Q^+\) denote the \(p\)-adic valuation of the integer \(n\), the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function \(f_p: \mathbb{N}^*\longrightarrow \mathbb{Q}^+\) defined by \(f_p(n):= (n/p^{v_p (n)})^{1 - v_p (n)}\) for any positive integer \(n\). We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the \(p\)-adic valuation. Some of these results extend Farhi's results from the case of even prime to that of odd prime.