OPEN
Let $\alpha,\beta\in (0,1)$. Does the density of integers $n$ such that the largest prime factor of $n$ is $<n^{\alpha}$ and the largest prime factor of $n+1$ is $<n^\beta$ exist?
Dickman
[Di30] showed the density of smooth $n$, with largest prime factor $<n^\alpha$, is $\rho(1/\alpha)$ where $\rho$ is the
Dickman function.
Erdős also asked whether infinitely many such $n$ even exist, but Meza has observed that this follows immediately from Schinzel's result [Sc67b] that for infinitely many $n$ the largest prime factor of $n(n+1)$ is at most $n^{O(1/\log\log n)}$.
