On solutions of ₈=₁ⁿ 1/xᵢ = 1 in integers of the form 2ᵃ kᵇ, where k is a fixed odd positive integer

We give an algorithm that produces all solutions of the equation ₈=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).