Show HN: Extending Axiom's PrimeGapsLib to prove parts 2 and 3 of Erdős #420

CollapseProblem 420

Statement

If τ(n)\tau(n)τ(n) counts the number of divisors of nnn then let

Is it true that F(log⁡n,n)F(\log n,n)F(logn,n) is everywhere dense in (1,∞)(1,\infty)(1,∞)?

More generally, if f(n)≤log⁡nf(n)\leq \log nf(n)≤logn is a monotonic function such that f(n)→∞f(n)\to \inftyf(n)→∞ as n→∞n\to \inftyn→∞, then is F(f,n)F(f,n)F(f,n) everywhere dense?

Status

Open

No claim settles this problem.

ReferencesEGIP96Erdős, Paul and Graham, S. W. and Ivić, Aleksandar and Pomerance, Carl, On the number of divisors of n!n!n!. (1996), 337-355.