On the Inferior and Superior k-th Power Part 0f a Positive Integer and Divisor Function

For any positive integer n, let a(n) and b(n) denote the inferior and superior k-th power part of n respectively. That is, a(n) denotes the largest k-th power less than or equal to n, and b(n) denotes the smallest k-th power greater than or equal to n. In this paper, we study the properties of the sequences{a(n)} and {b(n)}, and give two interesting asymptotic formulas.