  We establish new upper bounds for the length of runs of consecutive positive
integers each with exactly $k$ divisors, where $k$ is a given positive integer
of some special forms. Also we have found exact values of the maximum possible
runs for some fixed values of $k$. In addition, we exhibit the run of 11
consecutive positive integers each with exactly 36 divisors, the run of 13
consecutive positive integers each with exactly 12 divisors, the run of 17
consecutive positive integers each with exactly 24 divisors, and the run of 17
consecutive positive integers each with exactly 48 divisors.
