We give an algorithm that produces all solutions of the equation $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$, where $k$ is a fixed positive integer that is not a power of $2$, $a$ is an element of $\{0,1,2\}$ that can vary from term to term, and $b$ is a nonnegative integer that can vary from term to term. We also completely characterize the pairs $(k,n)$ for which this equation has a nontrivial solution in integers of this form.