Complete sequences of sets of integer powers

Call S complete if Σ(S) contains all sufficiently large integers. It has been known for some time (see [B]) that if gcd(a, b) = 1 then the (nondecreasing) sequence formed from the values ab with s0 ≤ s, t0 ≤ t ≤ f(s0, t0) is complete, where s0 and t0 are arbitrary, and f(s0, t0) is sufficiently large. In this note we consider the analogous question for sequences formed from pure powers of integers. Specifically, for a sequence A of integers greater than 1, denote by Pow(A; s) the (nondecreasing) sequence formed from all the powers a where a ∈ A and k ≥ s ≥ 1. Although we are currently unable to prove it, we believe the following should hold: