Runs of Consecutive Integers Having the Same Number of Divisors

Our objective is to provide an upper bound for the length ${\mathrm{\ell }}_N$ of the longest run of consecutive integers smaller than $N$ which have the same number of divisors. We prove in an elementary way that $\log {\mathrm{\ell }}_N\ll (\log N\log \log N{)}^{\lambda }$, where $\lambda =1\mathrm{/}2$. Using estimates for the Jacobsthal function, we then improve the result to $\lambda =1\mathrm{/}3$.