Is there a necessary and sufficient condition for a sequence of integers $b_1<b_2<\cdots$ that ensures there exists a primitive sequence $a_1<a_2<\cdots$ (i.e. no element divides another) with $a_n\ll b_n$ for all $n$?In particular, is this always possible if there are no non-trivial solutions to $(b_i,b_j)=b_k$?Similarly, find necessary and sufficient conditions on a sequence $n_1<n_2<\cdots$ that ensure there exists a primitive set $A$ such that$$|A\cap [1,2^{n_i}]|\gg 2^{n_i}$$for every $i$.