On solutions of ∑ᵢ₌₁ⁿ 1/xᵢ = 1 in integers of the form $2^a k^b$, where $k$ is a fixed odd positive integer

We give an algorithm that produces all solutions of the equation ∑ᵢ₌₁ⁿ 1/xᵢ = 1 in integers of the form 2ᵃ kᵇ, where k is a fixed odd positive integer at least $3$, 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 show that this equation has a nontrivial solution in integers of this form if and only if k ≤ 4n-11 or $(k,n) = (3,3)$.