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Claim. P. Erdős and E. G. Straus, Some number theoretic results, Pacific J. Math. 36 (1971), no. 3, 635--646, prove two results on the series

$$\sum_{n\ge1}\frac{d(n)}{a_1\cdots a_n}$$

of Problem 258, where $d(n)=\tau(n)$ is the number of divisors of $n$. Theorem 2.23 (printed p. 641) states that the series is irrational whenever $2\le a_1\le a_2\le\cdots$ is a monotonic sequence of integers; its proof joins two cases, Lemma 2.2 (some $\delta>0$ has $a_n<(\log n)^{1-\delta}$ for infinitely many $n$) and Lemma 2.14, whose proof uses the Dirichlet divisor theorem $\sum_{n\le N}d(n)\sim N\log N$, the almost-all bound $d(n)<(\log n)^{\log2+\varepsilon}$, and Lemma 2.17, which bounds $d(x+y)$ for almost all $x$ and every $y\ge3$. Lemma 2.14 (printed p. 640) states that if $|a_n|>c(\log n)^{3/4}$ for all $n$, with some constant $c>0$, then the series is irrational, and the paper adds that in this lemma "we need not assume the monotonicity of $a_n$" (p. 640), nor even that the $a_n$ are positive, the proof being written for positive $a_n$. The section's Conjecture 2.24 (printed p. 642) states the problem's question, irrationality for every sequence with $a_n\to\infty$, which Chojecki's deduction answers. The paper's Theorem 2.26 proves the analogous irrationality for $\sigma(n)$ and $\varphi(n)$ in place of $d(n)$ for monotonic integer sequences with $a_n\ge n^{11/12}$ for large $n$, the subject of the further conjecture the site's remarks record. The source card erdos_1971_number_theoretic_results names the author-archive scan (the second paper link) as the copy read; no file is held.

Covers. Two classes of sequences: every nondecreasing integer sequence with $a_1\ge2$ (Theorem 2.23), whether or not it tends to infinity, and every sequence of positive integers with $|a_n|>c(\log n)^{3/4}$ for all $n$ and some $c>0$ (Lemma 2.14), with no monotonicity. Not covered: sequences tending to infinity that are neither monotone nor bounded below by such a power of $\log n$, which is the gap the problem's question concerns.

Acceptance. Refereed: the Pacific Journal of Mathematics, volume 36, issue 3 (March 1971), pp. 635--646; the Crossref record of the DOI gives these data. The site labels the problem PROVED (LEAN) and credits Chojecki with the full answer; its remark that Erdős and Straus proved the monotone case credits this paper with a part only, so no reviewed evidence is listed. The proofs are not checked here.

Depends on. Nothing in this wiki.