Problem Statement
If $\tau(n)$ counts the number of divisors of $n$ then let\[F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.\]Is it true that\[\lim_{n\to \infty}F((\log n)^C,n)=\infty\]for large $C$?
Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$?
More generally, if $f(n)\leq \log n$ is a monotonic function such that $f(n)\to \infty$ as $n\to \infty$, then is $F(f,n)$ everywhere dense?
Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$?
More generally, if $f(n)\leq \log n$ is a monotonic function such that $f(n)\to \infty$ as $n\to \infty$, then is $F(f,n)$ everywhere dense?
Categories:
Number Theory
Progress
Erdős and Graham write that it is easy to show that $\lim F(n^{1/2},n)=\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.Erdős, Graham, Ivić, and Pomerance [EGIP96] have proved that\[\liminf F(c\log n, n) = 1\]for any $c>0$, and\[\lim F(n^{4/9},n)=\infty.\](The exponent $4/9$ can be improved slightly.) They also prove that, if $f(n)=o((\log n)^2)$, then for almost all $n$\[F(f,n)\sim 1.\]van Doorn notes in the comments that the existence of infinitely many bounded prime gaps implies\[\limsup_{n\to \infty}F(g(n),n)=\infty\]for any $g(n)\to \infty$, and that Cramér's conjecture implies\[\lim F(g(n)(\log n)^2, n)=\infty\]for any $g(n)\to \infty$>
Source: erdosproblems.com/420 | Last verified: January 15, 2026