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Claim. J. Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, INTEGERS 26 (2026), #A28, received 9 October 2025, accepted 15 January 2026, published 20 February 2026; the arXiv preprint of 8 April 2025 numbers its results differently. Corollary 1 (published numbering) states: a sequence $(a_n)_{n\ge1}$ of positive integers with

$$\frac23\le\frac{a_n^2}{a_{n+1}}\le\frac43\quad\text{for every }n \qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac1{a_n}=1$$

is Sylvester's sequence $2,3,7,43,\ldots$, so $a_{n+1}=a_n^2-a_n+1$ for every $n$. It follows from Theorem 1, which shows that sequences whose ratios $a_n^2/a_{n+1}$ lie in a fixed range are almost determined by their reciprocal sums. Theorem 3 states that Question 1 of the paper, the question of Problem 243, has an affirmative answer if and only if the paper's Conjecture 1 holds: for a positive rational $r$, if the gap sequence $\varepsilon_n$ of the pseudo-greedy expansion of $r$ (the unit-fraction expansion whose $n$th term is the integer nearest to $1/(r-\sum_{k<n}1/a_k)$, plus one) tends to $0$, then $\varepsilon_n=0$ for all large $n$. The author reports a computer check of Conjecture 1 for $r=p/q$ with $0<p\le q\le10^5$; this equivalence and check are a reduction and evidence, not a settled part of the problem. Corollary 4, attributed to Badea and to Erdős and Straus, recovers within the same framework the cases where $a_{n+1}\ge a_n^2-a_n+1$ for all large $n$ or $\liminf(a_1\cdots a_{n-1}/a_n)(1-a_n^2/a_{n+1})\ge0$. V. Kovač pointed the paper out on the site's discussion thread in a comment of 11 September 2025, in reply to T. Tao's comment giving the same observations; Tao then recorded that the paper already contains them. The source card koizumi_2025_irrationality_reciprocal_sum_doubly_exponential_sequences digests the arXiv version, where the equivalence is Theorem 16 (p. 10) and the conjecture is Conjecture 6.

Covers. The sequences of Problem 243 with $2/3\le a_n^2/a_{n+1}\le4/3$ for every $n$ and reciprocal sum exactly $1$: each is Sylvester's sequence, so the recurrence holds from the first term. Not covered: other rational sums, and sequences whose ratios leave the interval for some $n$.

Acceptance. Refereed: INTEGERS 26 (2026), #A28, with the journal's received, accepted and published dates. The site labels the problem OPEN and does not credit the paper, so no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.