## Sequences with bounded lcm for consecutive elements

On page 83 of the pdf-file of Old and New Problems and Results in Combinatorial Number Theory, the following is asked: If we have an infinite sequence $1 \le a_1 < a_2 < ..$ of positive integers and set $A(n)$ to be the number of indices $i$ such that $\text{lcm}(a_i, a_{i+1}) \le n$, do we then have $A(n) = O(\sqrt{n})$? In the following very short article I answer the finite version of this question, which also implies an affirmative answer to the original question: Sequences with bounded lcm for consecutive elements . I conjecture that the term $\log(2n)$ is superfluous, but to get rid of it completely, a more detailed approach is needed. By the way, the constant $c \approx 1.86$ is best possible, in the sense that it is possible to construct an infinite sequence such that, for every $\epsilon > 0$ and infinitely many $n$, we have $A(n) > (c-\epsilon)\sqrt{n}$.