Consecutive integers divisible by the number of their divisors

Abstract: 

In 1985, Spiro estimated the counting function of the set of positive integers nn which are divisible by the number of their divisors. In my talk, we will look at consecutive integers each divisible by the number of its divisors. We show that there are no strings of 33 such consecutive integers, and that the counting function of the set of n≤xn≤x such both nn and n+1n+1 are multiples of the number of their respective divisors is x1/2(loglogx)O(1)/(logx)cx1/2(log⁡log⁡x)O(1)/(log⁡x)c, where c=2−1/3–√c=2−1/3.