On sets of numbers recognized by push-down automat

A set of positive integers is said to be recognizable by a push-down automaton if its elements, written in k-ary notation for some k ≥ 2, form a context-free language. Some general properties of this type of sets are given. The set of squares is shown not to be recognizable. A necessary condition is proved for a subset of a set defined by a linear recurrence relation of some special form to be recognizable. The set of square-free integers is investigated.