OPEN
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 $(b_i,b_j)=b_k$?
A problem of Erdős, Sárközy, and Szemerédi
[ESS68]. 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. (The former is due to Erdős
[Er35], the latter to Erdős, Sárközy, and Szemerédi
[ESS67].)
One can ask a similar question for sequences of real numbers, as in [143].
