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Problem 1184 — Erdős Problems

Problem 1184

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?

Source: erdosproblems.com/1184

Number theory
Primes

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Er76e: Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
RST75b: Ramachandra, K. and Shorey, T. N. and Tijdeman, R., On Grimm's problem relating to factorisation of a block of consecutive integers. II. J. Reine Angew. Math. (1976), 192-201.

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