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

摘要

Erdo{double acute}s, Freud and Hegyvári [1] constructed a permutation a_1, a_2,... of positive integers with [a_i, a_(i+1)] < i exp {c√log i log log i} for an absolute constant c ge; 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 i0 with [a_i, a_(i+1)] < i exp {(2√2 + ε)√ log i log log i} for all i ≧ i_0.