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Claim. The answer to Problem 1099 is yes, along the explicit sequences Erdős proposed. Gérald Tenenbaum, Sur un problème extrémal en arithmétique, Annales de l'Institut Fourier 37 (1987), no. 2, 1--18, studies

$$F(N;h)=\sum_{1\le i<\tau(N)}h\!\left(\frac{d_{i+1}}{d_i}-1\right)$$

for $h$ in the class of continuously differentiable $h:[0,\infty)\to[0,\infty)$ with $h(0)=h'(0)=0$ for which $(1+u)h'(u)$ is differentiable and nondecreasing on $u>0$. Its Théorème 1 states that if a convergence condition on $h$, the paper's (3), holds for some $\beta$ with $0<\beta<3/2$, then $F(N;h)=O(1)$ uniformly for $N$ in the factorials $k!$, the least common multiples $\mathrm{lcm}(1,\dots,k)$ and the primorials $p_1\cdots p_k$ together with $1$. The paper then notes that $h(u)=u^{\alpha}$ lies in the class for every $\alpha>1$ and satisfies the condition for every $\beta>1$, so that

$$h_\alpha(N)=O_\alpha(1)$$

along all three sequences, which it calls the second part of Erdős's conjecture; in particular $\liminf_n h_\alpha(n)\ll_\alpha1$. The paper attributes the reduction it builds on, its Théorème 2, to Vose, and derives Théorème 1 from it by exhibiting admissible sequences for the three families. The site's commentary credits Vose with the main question and reports the factorial and least-common-multiple cases as unsettled; this theorem settles them. The journal record gives the year and issue and no day, so this page carries the first of January.

Corrigendum. A corrigendum, Annales de l'Institut Fourier 50 (2000), no. 1, 317--319 (the second paper link), says that the proof of Lemme 3 is incorrect because its formula (18) does not follow from what precedes it, and replaces that lemma by a Lemme 3' that repairs the proof of Théorème 3, on which the admissibility of the paper's (5), and so Théorème 1, rests; it also corrects a misprint in the statement of Théorème 4. The statement of Théorème 1 is unchanged.

Depends on. No page of this wiki.

Acceptance. The paper is a refereed article in the Annales de l'Institut Fourier, the refereed evidence. The site's problem page credits Vose alone; a comment in the problem's thread on 2 July 2026 points out that Tenenbaum's paper also solves the problem, and no curator credit of it is recorded, so no reviewed evidence is listed. No proof has been reproduced here.