# Consecutive integers divisible by the number of their divisors

Florian Luca

## Affiliation:

University of the Witwatersrand

## Date:

Wed, 03/02/2016 - 2:30pm

Red Centre M032

## Abstract:

In 1985, Spiro estimated the counting function of the set of positive integers $n$ 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 $3$ such consecutive integers, and that the counting function of the set of $n\le x$ such both $n$ and $n+1$ are multiples of the number of their respective divisors is $x^{1/2}(\log\log x)^{O(1)}/(\log x)^c$, where $c=2-1/{\sqrt{3}}$.