Density and growth of $\tau((n+f(n))!)/\tau(n!)$, ratios of divisor-counts of nearby factorials (Erdős #420)
Statement
For a positive integer $n$ let $\tau(n)$ be the number of divisors of $n$, and for a function $f$ set $$F(f,n)=\frac{\tau\big((n+\lfloor f(n)\rfloor)!\big)}{\tau(n!)},$$ the ratio of the divisor-count of $(n+\lfloor f(n)\rfloor)!$ to that of $n!$. Three questions are asked. (1) For large constants $C$, is $\lim_{n\to\infty}F((\log n)^C,n)=\infty$? (2) Is $F(\log n, n)$ everywhere dense in $(1,\infty)$? (3) More generally, if $f(n)\leq \log n$ is a monotonic function with $f(n)\to\infty$ as $n\to\infty$, is $F(f,n)$ everywhere dense in $(1,\infty)$?
Acceptance. FULLY RESOLVES: complete proofs (written in full, or machine-checkable) settling the three questions — establishing or refuting $\lim_{n\to\infty}F((\log n)^C,n)=\infty$ for large $C$, and deciding the everywhere-density in $(1,\infty)$ of $F(\log n,n)$ and of $F(f,n)$ for monotone $f\leq\log n$ with $f\to\infty$. ADVANCES: (a) an unconditional proof of any one of the three questions; (b) a strict improvement of a bound stated in the background — e.g. lowering the exponent $4/9$ in '$\lim F(n^{\alpha},n)=\infty$' below the best value stated here, or extending the almost-all $F(f,n)\sim 1$ range beyond $f(n)=o((\log n)^2)$ — with full proof; (c) making one of van Doorn's conditional implications unconditional; (d) a large-scale computation of $F(f,n)$ using exact $\tau(n!)$ that convincingly maps the density/limit behaviour, delivered with reproducible code and labelled as evidence. Deliver the proof file, or the computation plus code.
Background
An Erdős–Graham problem [ErGr80, p.83]; listed as open on erdosproblems.com/420 (fetched 2026-07-13, status 'open', tagged 'number theory'). The quantity $\tau(n!)$ is fully explicit: by Legendre's formula the exponent of a prime $p$ in $n!$ is $\sum_{i\geq 1}\lfloor n/p^i\rfloor$, so $\tau(n!)=\prod_{p\leq n}\big(1+\sum_{i\geq 1}\lfloor n/p^i\rfloor\big)$, and $F(f,n)$ measures how much appending $\lfloor f(n)\rfloor$ further factors inflates this divisor count. Frontier: Erdős and Graham noted it is easy that $\lim F(n^{1/2},n)=\infty$, with $n^{1/2}$ replaceable by $n^{1/2-c}$ for a small $c>0$. Erdős, Graham, Ivić and Pomerance [EGIP96] proved $\liminf F(c\log n,n)=1$ for every $c>0$ and $\lim F(n^{4/9},n)=\infty$ (the exponent $4/9$ improvable slightly), and that if $f(n)=o((\log n)^2)$ then $F(f,n)\sim 1$ for almost all $n$. In the site comments van Doorn observes that the existence of infinitely many bounded prime gaps implies $\limsup_n F(g(n),n)=\infty$ for any $g(n)\to\infty$, and that Cramér's conjecture implies $\lim F(g(n)(\log n)^2,n)=\infty$ for any $g(n)\to\infty$. No cash prize is attached. The attacker's tool: exact computation of $\tau(n!)$ via Legendre's formula lets one evaluate $F(f,n)$ over large ranges to probe the density and limit questions empirically; the proof frontier is analytic, blending the EGIP96 divisor-in-factorial estimates with results on gaps between primes (bounded gaps, Cramér-type heuristics).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #420 (T. F. Bloom) | website |
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