OPEN
This is open, and cannot be resolved with a finite computation.
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].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #892, https://www.erdosproblems.com/892, accessed 2025-11-16
