Abstract
In this paper we study the conditions (1), (2) and (3) for the permutations which preserve the weighted density. These conditions are motivated by the conditions of Lévy group, originated in [Levy, P.: Problèmes concrets d’Analyse Fonctionelle. Gauthier Villars, Paris, 1951], and studied in [Obata, N.: Density of natural numbers and Lévy group, J. Number Theory 30 (1988), 288-297]. In the second part we prove that under some conditions for the sequence of weights, for instance for the logarithmic density, the first two conditions can be launched