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on Erdős #1184 · number theory / primes

The statement

Let $f(n,k)$ count the number of $1\leq i\leq k$ 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?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.