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}.

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