An upper bound of Σ 1/(ai log ai) for quasi-primitive sequences

A nonempty sequence A = {ai} of integers a1 < a2 < …, which are greater than or equal to two, is called quasi-primitive if there do not exist three distinct integers ai, aj, ak A such that (ai, aj) = ak. We show that Σ 1/(ai log ai < 4.2022 for any quasi-primitive sequence.