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Claim. Let $S(N)$ be the number of distinct values of $\sum_{k\le N}\varepsilon_k/k$ with $\varepsilon_k\in\{0,1\}$ and let $\log_j$ be the $j$-fold iterated natural logarithm. For $r\ge1$ and $\log_{2r}N\ge1$,

$$\log S(N)\ \le\ \frac{N\log_rN}{\log N}\prod_{j=3}^{r}\log_jN .$$

This is Theorem 3 of the paper (p. 610), in the form of its introduction (p. 598) and of the site's page for Problem 320. The proof splits $\{1,\ldots,N\}$ by the presence of a prime factor above $N/\log N$ and recurses on $r$. The same paper's Theorem 2 (p. 603) gives the lower bound $\log S(N)\ge\frac1e\frac{N}{\log N}\prod_{j=3}^r\log_jN$ under the same condition; it is weaker than the 1975 bound of Bleicher and Erdős's Corollary 3 and is recorded here.

Covers. An upper bound for $\log S(N)$. The condition $\log_{2r}N\ge1$ allows a depth $r$ of at most about half the depth at which the iterated logarithm becomes bounded, and the extra factor $\log_rN$ is then a tower over the remaining iterated logarithms, so the bound exceeds the order of magnitude $\frac{N}{\log N}\prod_{j=3}^k\log_jN$ with $\log_kN=O(1)$ by an unbounded factor. The upper bound of that order is Young, Zhu and Luo's accepted claim.

Depends on. No page of this wiki; the theorem rests on the paper's own lemmas.

Acceptance. Refereed: M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), no. 4, 598--613, DOI 10.1215/ijm/1256049650; the issue is dated 1 December 1976 in the Crossref record, the date this page carries. The proofs are not verified by this corpus.