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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 m0 and n0 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(pm0qn0) has, at most, one solution of the form x = pmqn such that pmqn = pm0qn0.

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