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Claim. The answer to Problem 1099 is yes. Michael D. Vose, Integers with consecutive divisors in small ratio, Journal of Number Theory 19 (1984), no. 2, 233--238, constructs an infinite sequence of integers $N$ with

$$h_\alpha(N)=\sum_i\left(\frac{d_{i+1}}{d_i}-1\right)^{\alpha}=O_\alpha(1) \qquad(\alpha>1),$$

so that $\liminf_{n\to\infty}h_\alpha(n)\ll_\alpha1$; the term $i=1$ shows that the limit inferior is at least $1$, so the bound is of the right shape. Tenenbaum's 1987 account of the paper describes its opening step: it suffices to build a sequence $N_1=1$, $N_k\mid N_{k+1}$ and to impose the closeness condition only on the divisors of $N_k$ lying in $(\sqrt{N_{k-1}},\sqrt{N_k}]$. Vose's construction does not reach the factorials or the least common multiples $\mathrm{lcm}(1,\dots,k)$ that Erdős proposed as candidates; those sequences are covered by Tenenbaum's claim page. The journal issue is dated October 1984 and no day is recorded, so this page carries the first of that month. The theorem is stated here as Tenenbaum's introduction (p. 2) and the site's credit state it.

Depends on. No page of this wiki.

Acceptance. Thomas Bloom, the site's curator, labels the problem proved and credits Vose's paper on the problem page, last edited 19 October 2025, after the discussion of that day in which the paper was located and Woett confirmed that it settles the main question; that credit is the reviewed evidence. The paper is a refereed article in the Journal of Number Theory, the refereed evidence. No proof has been reproduced here.

Formalization. The Lean file linked above, Erdos1099.lean in Boris Alexeev's repository of Lean proofs, pinned to the commit the formal-conjectures statement file cites, declares itself a formalization of a solution to the problem with Vose as its informal author and Codex and GPT-5.6 Sol as its formal authors; its theorem erdos_1099 states that for every real $\alpha>1$ the sum $h_\alpha$ is bounded on an infinite set of integers. The formal-conjectures statement for the problem has pointed to it as a formal proof since 20 September 2026. This corpus has not built the file, so no formalized evidence is listed.