An abundancy result for the two prime power case and results for an equations of Goormaghtigh

Let $a$ represent a positive integer, and $sigma (a)$ denote the sum of the (positive) factors of $a$. The emph{abundancy} (or emph{abundancy index}) of $a$, denoted by $I(a)$, is defined by $I(a) = sigma (a)/a$. Suppose that we are given positive integers $m_0$ and $n_0$ and distinct prime numbers $p$ and $q$. In the current note, new results concerning an equation of Goormaghtigh are established and used to show that the equation $I(x) = I(p^{m_0} q^{n_0})$ has, at most, one solution of the form $ x = p^{m} q^{n}$ such that $p^{m} q^{n} ne p^{m_0} q^{n_0}$.