Abstract: A principal divisor of an integer is a maximal prime-power divisor, that is, if p is prime and a is a positive integer, then p^a is a principal divisor of n if p^a is a divisor of n but p^(a+1) is not a divisor of n. Thus, the number of principal divisors of n is equal to the number of distinct prime divisors of n. Let P_k be the set of all positive integers with exactly k principal divisors. Then
P_1 = {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...},
P_2 = {6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, ...},
P_3 = {30, 42, 60, 66, 70, 78, 90, 102, 105, 110, ...}, etc.
There are some obvious runs of consecutive integers in P_1, such as the run of size 4 starting with 2, and the run of size 3 starting with 7. We denote them by [2, 4] and [7, 3] , where [a, r] denotes the maximal run of size r with starter a. Beyond [7, 3] the next nontrivial runs are [16, 2] and [31, 2]. In fact, it is easy to see that after [7, 3] there is no later run of size bigger than 2 in P_1, but no-one knows whether there are infinitely many runs of size 2 in P_1. Some obvious runs of consecutive integers in P_2 are [14, 2], [20, 3] and [33, 4]. How large can runs in P_2 get? That is our main topic for this talk. We shall also discuss nontrivial runs in P_k when k > 2.