On the Sum of Unitary Divisors Maximum Function

It is well-known that a positive integer $d$ is called a unitary divisor of an integer $n$ if $d|n$ and gcd$\left(d,\frac{n}{d}\right)=1$. Divisor function $\sigma^{*}(n)$ denote the sum of all such unitary divisors of $n$. In this paper we consider the maximum function $U^{*}(n)=\max\{k\in\mathbb{N}:\sigma^{*}(k)|n\}$and study the function $U^{*}(n)$ for $n=p^{m}$, where $p$ is a prime and $m\geq 1$.