Further Results on a Curious Arithmetic Function

الملخص EN

Let p be an odd prime number and n be a positive integer.

Let vpn, N∗, and 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 fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn 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.