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

Erd?s, Freud and Hegyvári 1] constructed a permutation a ₁,a ₂,… of positive integers with \(a_{i}, a_{i+1}]< i\exp \left\{c\sqrt{\log i}\log\log i\,\right\}\) for an absolute constant c>0 and all i≧3. In this note, we construct a permutation of all positive integers such that for any ε>0 there exists an i ₀ with \(a_{i}, a_{i+1}]\allowbreak < i\exp \left\{\left(2\sqrt{2}+\varepsilon\right) \sqrt{\log i\log\log i}\,\right\}\) for all i≧i ₀.