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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 $\log_j$ be the $j$-fold iterated natural logarithm. Let $\mathcal Q(N)$ be the set of $n\le N$ that are products $p_1\cdots p_k$ of primes with $p_i>e^{3p_{i-1}/2}$, over all $k$, and $\mathcal Q_k(N)$ its members with exactly $k$ prime factors. The Lemma of p. 40 says that two sequences of distinct elements of $\mathcal Q(N)$ have equal reciprocal sums only if they coincide up to order, so $\mathcal Q(N)$ has distinct subset reciprocal sums, and the theorem of p. 30 bounds $Q_k(N)=|\mathcal Q_k(N)|$ below. Hence, for $k\ge3$ and $\log_{k+1}N\ge k+1$,

$$R(N)\ \ge\ |\mathcal Q(N)|\ \ge\ Q_k(N)\ \ge\ \frac{N}{\log N}\prod_{j=3}^{k+1}\log_jN,$$

which is the lower bound the site prints for Problem 321 with $k+1$ renamed $k$, and the one the 1980 monograph attributes to this paper. The paper itself uses the Lemma to prove $S(N)\ge2^{Q(N)}$ (pp. 39--42) and does not state the consequence for $R(N)$.

Covers. A lower bound for $R(N)$. The condition $\log_{k+1}N\ge k+1$ stops the product at a depth where the iterated logarithm is still large, so the bound falls short of the order of magnitude by an unbounded factor; the bound of that order is the set of Bettin, Grenié, Molteni and Sanna.

Depends on. The theorem of p. 39 and its Lemma and the theorem of p. 30, the library pages that state the Lemma and the count.

Acceptance. Refereed: M. N. Bleicher and P. Erdős, The number of distinct subsums of $\sum_1^N1/i$, Math. Comp. 29 (1975), no. 129, 29--42, DOI 10.1090/S0025-5718-1975-0366795-4. The proofs are not verified by this corpus.

Dating. The page is dated by the publication year; the Crossref record gives the year only, and the day in the page name is a placeholder.