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Claim. Let $R(N)$ be the largest size of a set $A\subseteq\{1,\ldots,N\}$ whose subset sums $\sum_{n\in S}1/n$, $S\subseteq A$, are pairwise distinct, and let $\ln_j$ be the $j$-fold iterated natural logarithm. Let $\mathcal U$ be the set of integers $u\ge1$ such that $1/u$ is not a $\{-1,0,1\}$-combination of $1/1,\ldots,1/(u-1)$, and $\mathcal U(N)=\mathcal U\cap[1,N]$. The proof of Theorem 1 of the paper (arXiv v1, p. 9) bounds $|\mathcal U(N)|$ below by $2(1-\frac{1.4}{\ln_kN})\frac{N}{\ln N}\prod_{j=3}^k\ln_jN$ for $k\ge4$, and its Lemma 2 gives $S(N)\ge2^{|\mathcal U(N)|}$, where $S(N)$ counts the distinct reciprocal subset sums of $\{1,\ldots,N\}$. The paper does not state that $\mathcal U(N)$ has distinct subset reciprocal sums; that step is the three-line deduction on the theorem's library page: if two distinct subsets had equal sums, dropping their common elements and taking the largest remaining element $u$ would write $1/u$ as a $\{-1,0,1\}$-combination of smaller reciprocals. Hence, for $k\ge4$ and $\ln_kN\ge3/2$,

$$R(N)\ \ge\ |\mathcal U(N)|\ \ge\ 2\Bigl(1-\frac{3/2}{\ln_kN}\Bigr)\frac{N}{\ln N}\prod_{j=3}^{k}\ln_jN,$$

a bound of the order $\frac{N}{\log N}\prod_{j=3}^k\log_jN$ with $\log_kN=O(1)$ for Problem 321.

Covers. The lower half of the order of magnitude of $R(N)$. Not covered: the upper bound.

Depends on. Theorem 1 and its relation to Problem 321, the library page that states the theorem, the proof's bound for $|\mathcal U(N)|$ and the deduction that $\mathcal U(N)$ is dissociated.

Acceptance. Refereed: the paper appeared in Mathematics of Computation, DOI 10.1090/mcom/4190, published online 22 January 2026; the page is dated by the arXiv posting of 12 September 2025. Reviewed: the site's curator, Thomas Bloom, labels the problem SOLVED for the order of magnitude and in its commentary calls the lower bound implicit in this work; Bloom's comment of 16 July 2026 under the accepted claim on the tab says that the lower bound comes from that earlier work. The curator is independent of the authors. The proof is not verified by this corpus.