Divisibility of integers obtained from truncated periodic sequences

A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]