Let $f(n,k)$ count the number of $1\le i\le k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\alpha >1$ is such that $n=k^{\alpha +o(1)}$, then$$f(n,k)=(1-\rho (\alpha )+o(1))k,$$where $\rho$ is the Dickman function?