Square-free and square-full part of an integer

Abstract

Any positive integer [Formula: see text] can be written uniquely as [Formula: see text] where [Formula: see text] is square-full, [Formula: see text] is square-free and [Formula: see text] We denote the square-full part and the square-free part of [Formula: see text] by [Formula: see text] and [Formula: see text] respectively. In this paper, we use an elementary method to study the distribution of square-free and square-full part of an integer. Moreover, we also study the distribution of primitive roots modulo a prime [Formula: see text] that are square-full number.