Problem 892

Erdős Problem #892: declared status 'open'. Formalized: no. 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 $(b_i,b_j)=b_k$? Current best: It is known that$$\sum \frac{1}{b_n\log b_n}<\infty$$and$$\sum_{b_n<x}\frac{1}{b_n} =o\left(\frac{\log x}{\sqrt{\log\log x}}\right)$$are both necessary. Prize: no. Tags: number theory, primitive sets.