A question of Erdős, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that $F(n)\leq n$ for only finitely many $n$, and in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$ (note that it is trivially $\leq n+\sqrt{n}$).

Tao has discussed this problem in a blog post.

Sarosh Adenwalla has observed that the first question is equivalent to [430]. Indeed, if $n$ is large and $a_i$ is the sequence defined in the latter problem, then [430] implies that there is a composite $a_j$ such that $a_j-p(a_j)>n$ and hence $F(n)>n$.

See also [463].

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