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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 $k\ge4$ and $\log_kN\ge k$,

$$\log S(N)\ \ge\ \frac{N\log2}{\log N}\prod_{j=3}^{k}\log_jN .$$

This is Corollary 3 of the paper (p. 40), which states it for $k\ge3$ and $\log_{k+1}N\ge k+1$ with the product to $k+1$; the form above, with $k+1$ renamed $k$, is the one the site prints for Problem 320. The paper derives it from its theorem $S(N)\ge2^{Q(N)}$ (p. 39) and its count of the integers $p_1\cdots p_k\le N$ built from rapidly growing primes (p. 30).

Covers. A lower bound for $\log S(N)$. The condition $\log_kN\ge k$ stops the product at a depth where $\log_kN$ is still at least $k$, so the bound falls short of the order of magnitude $\frac{N}{\log N}\prod_{j=3}^k\log_jN$ with $\log_kN=O(1)$ by an unbounded factor; the bound of that order is Bettin, Grenié, Molteni and Sanna's Theorem 1.

Depends on. No page of this wiki; the corollary rests on the paper's own theorems.

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.