$d$-complete sequences of integers

An infinite sequence $a_1<a_2<\dotsb$ is $d$-complete if every sufficiently large integer is the sum of $a_i$ such that no one divides the other. We investigate $d$-completeness of sets of the form $\{p^\alpha q^\beta\}$ and $\{p^\alpha q^\beta r^\gamma\}$ with $\alpha,\beta,\gamma$ nonnegative.