On strings of consecutive integers with a distinct number of prime factors

Abstract:

Let $\omega (n)$ be the number of distinct prime factors of $n$. For any positive integer $k$ let $n=n_k$ be the smallest positive integer such that $\omega (n+1),\ldots ,\omega (n+k)$ are mutually distinct. In this paper, we give upper and lower bounds for $n_k$. We study the same quantity when $\omega (n)$ is replaced by $\Omega (n)$, the total number of prime factors of $n$ counted with repetitions.