../

---

Source. Wouter van Doorn and GPT-6 Astra Pro (the author line as printed), Practical numbers and Egyptian fractions, Theorem 1.1, physical p. 1, deduced from Proposition 4.1 on pp. 5–6 (Section 4 opens by calling the proposition the stronger version of the theorem), in the seven-page PDF held by van Doorn (2026); the library records it on its result page. Read on the page images. Uses the Proposition 4.1 reconstruction.

Standing. Author-recorded reconstruction of a claimed result: the note is a proof claim on the erdosproblems.com proof-claims tab of Problem 18 (registered as partial because it answers only the first of the three questions), mostly AI-generated by its own account, not refereed, not on arXiv, with an author-side Lean formalization that was not built here; the site shows OPEN and no independent acceptance is documented. This page is not an independent review; it changes no status and assigns no tier. The chain of claimed results reconstructed in this folder is Lemma 3.1, Lemma 3.2, Lemma 3.3, Corollary 3.4, Proposition 4.1, then this theorem; the reading found no gap in the chain beyond the imported inputs named on each page, which is a reading, not a review.

Statement

Let $c_0=14/\log2$, all logarithms being natural (the note fixes this at the end of its Section 1, physical p. 2). There are infinitely many practical numbers $n$ with

$$h(n)\le c_0(\log\log n)^2 ,$$

where $h(n)$ is the least integer such that every positive integer $m\le n$ is a sum of at most $h(n)$ distinct divisors of $n$.

Proof

Let $E$ and $x_0$ be as in Proposition 4.1, and fix any odd prime $p_*$, say $p_*=3$. For each real $x\ge x_0$ the proposition gives a practical $n$ with $x\le n<x^2$ and $h(n)\le c_0(\log\log x)^2-1<c_0(\log\log n)^2$. Taking $x=x_0,x_0^2,x_0^4,\dots$ produces practical numbers $n$ in the disjoint intervals $[x_0^{2^i},x_0^{2^{i+1}})$, hence infinitely many distinct ones, each satisfying the bound.

What it would settle

The first question of Problem 18 asks for infinitely many practical $m$ with $h(m)<(\log\log m)^{O(1)}$. The site's definition ranges over $1\le m'<m$ and the note's over $m'\le m$; they differ only by the one-divisor representation of $m$ itself, so the note's $h$ is at least the site's. If the claim is correct it answers the first question with exponent $2$ and the explicit constant $c_0\approx20.2$, improving the $(\log m)^{1/2}$ of Vose's construction; it says nothing about $h(n!)$, the subject of the second and third questions.