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Updated Oct 7

Problem 420

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?

Status. Open.

Source. erdosproblems.com/420, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #420, https://www.erdosproblems.com/420.

References.

[EGIP96] Erdős, Paul and Graham, S. W. and Ivić, Aleksandar and Pomerance, Carl, On the number of divisors of $n!$ . (1996), 337-355.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (11)

Arithmetic Functions Problem 419 Arithmetic Functions factorials_binomials/erdos_1996_number_divisors Corollary 2 (p. 6): K(n) > log n log log n log log log log n/(9 (log log log n)^3) infinitely often Corollary 3 (p. 9): K(n) < n^{4/9} for all sufficiently large n Lemma 1 (p. 3): 1 + S(n)/2n <= d(n!)/d((n-1)!) <= 1 + 2S(n)/n Lemma 3 (p. 8): ≫ x^{4/9} primes above x^{5/9+δ} divide integers in (x, x + x^{4/9}]Theorem 1 (p. 3): an asymptotic expansion of log d(n!) in powers of 1/log n Theorem 3 (p. 5): f(n) >= (1/4 - ε) log n log log n log log log log n/(log log log n)^3 infinitely often Theorem 4 (p. 8): f(n) < n^{4/9} for all sufficiently large n

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