This function was considered by de Bruijn, Erdős, and Turán, who showed that\[\sum_{n<x}f(n)\sim \sum_{n<x}f(n)^2\sim x.\]The existence of some $c>0$ such that there are $\gg n^c/\log n$ many primes in $[n,n+n^c]$ implies that $\liminf f(n)>0$.

Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\epsilon>0$, if $x$ is sufficiently large there exists $y<x$ such that\[\pi(x)< \pi(y)+\epsilon \pi(x-y).\](Compare this to [855].) He notes that if\[\pi(x)< \pi(y)+O\left(\frac{x-y}{\log x}\right)\]for all $y<x-(\log x)^C$ for some constant $C>0$ then $f(n)\ll \log\log\log n$.

The study of $f(p)$ is even harder, and Erdős could not prove that\[\sum_{p<x}f(p)^2\sim \pi(x).\]

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