Problem 381

../

claims/: The 2 claim pages of Problem 381, one per claimant's result; the problem's standing derives from them.

---

Statement. A number $n$ is highly composite if $\tau(m)<\tau(n)$ for all $m<n$, where $\tau(m)$ counts the number of divisors of $m$. Let $Q(x)$ count the number of highly composite numbers in $[1,x]$.

Is it true that

$$Q(x)\gg_k (\log x)^k$$

for every $k\geq 1$?

Status. Disproved. The site's label; Nicolas's 1971 upper bound $Q(x)\ll(\log x)^{1+c'}$ answers the question no, as the claim page below records.

Source. erdosproblems.com/381, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #381, https://www.erdosproblems.com/381.

References.

• [Er44] Erdős, P., On highly composite numbers. J. London Math. Soc. (1944), 130-133.
• [Ni71] Nicolas, Jean-Louis, Répartition des nombres hautement composés de Ramanujan. Canadian J. Math. (1971), 116-130.

Formalization. None recorded.

Current assessment

The question is the site's formulation, unchanged between 2026-09-04 and 2026-10-07: with $Q(x)$ the number of highly composite numbers in $[1,x]$, whether $Q(x)\gg_k(\log x)^k$ for every $k\ge1$. The answer is no.

Erdős [Er44] proved $Q(x)>(\log x)^{1+c}$ for some $c>0$, through the gap bound that the highly composite number after $n$ is below $n+n(\log n)^{-c}$, and wrote that he could not decide whether every power of $\log x$ is exceeded; that is this question (card). The bound is the accepted partial claim Erdős 1944. Nicolas [Ni71], Théorème 4, proved the upper bound $Q(x)=O((\log x)^{1+c'})$, with another constant $c'$, by counting the highly composite numbers between consecutive superior highly composite numbers, so $Q(x)$ stays below a fixed power of $\log x$ (card). The claim page Nicolas 1971 records the result, its refereed venue and the curator's credit, and the problem's standing derives from it.

What remains is the exact growth: both bounds are powers of $\log x$ with unknown exponents, and Nicolas conjectures $\log Q(x)/\log\log x\to\log30/\log16=1.2267\ldots$; no later result on the exponent is recorded here. No formalization is recorded, and this repository has not checked Nicolas's proof independently; the account rests on the site page and the two cited papers' publication records.

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.

• erdos_1944_highly_composite_numbers
• erdos_1944_highly_composite_numbers / question_p130
• erdos_1944_highly_composite_numbers / theorem
• nicolas_1971_repartition_des_nombres_hautement_composes
• nicolas_1971_repartition_des_nombres_hautement_composes / conjecture_p117
• nicolas_1971_repartition_des_nombres_hautement_composes / theorem_3
• nicolas_1971_repartition_des_nombres_hautement_composes / theorem_4
• nicolas_1971_repartition_des_nombres_hautement_composes / theorem_5