Problem Statement
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 $(b_i,b_j)=b_k$?
In particular, is this always possible if there are no non-trivial solutions to $(b_i,b_j)=b_k$?
Categories:
Number Theory Primitive Sets
Progress
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].
Source: erdosproblems.com/892 | Last verified: January 19, 2026