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