Let a represent a positive integer, and σ(a) denote the sum of the (positive) factors of a. The abundancy (or abundancy index ) of a, denoted by I(a), is defined by I(a) = σ(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 = p m 0 q n 0 .