On the distribution of consecutive

Let M be a set of positive integers with its minimal element m ≥ 2. A positive integer n is called an M-free integer if every exponent of its prime factors is not contained in M. In this paper, we derive a weak asymptotic formula for the consecutive M-free integers by an elementary method. Some new results of this problem for special integers ((k, r)-integer, (k,r)-free integer, semi-r-free integer, unitarily r-free integer, exponentially odd integer, and almost exponentially even integer) are obtained.