The permutation of integers with small least common multiple of two subsequent terms (Q653948)

Improving a result of \textit{P. Erdős, R. Freud} and \textit{N. Hegyvári} [Acta Math. Hung. 41, 169--176 (1983; Zbl 0518.10063)], the authors construct an infinite permutation \(a_1,a_2,\ldots\) of all positive integers with the property that for every \(\varepsilon > 0\) there exists a positive integer \(i_0\) such that \[ [a_i, a_{i+1}] < i\exp \left((2\sqrt{2}+\varepsilon)\sqrt{\log i \log \log i} \right) \] holds for every \(i\geq i_0\). Here \([a_i, a_{i+1}]\) stands for the least common multiple of the consecutive terms \(a_i\) and \(a_{i+1}\).