We bound from above the length of the longest sequence of consecutive numbers less than or equal to x with the same number of divisors. We also bound the length of the longest sequence of consecutive numbers less than or equal to x for which the number of divisors is decreasing. Finally, we consider variants of this problem such as the corresponding sequences for the sum-of-proper-divisors function and the Carmichael function. In particular, we show that it is impossible for the sum-of-proper-divisors function to be equal on six consecutive integers.