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Claim. Let $F(N)$ be the number of positive integers of the form $1/n_1+\cdots+1/n_k$ with $1\le n_1<\cdots<n_k\le N$, the count of Problem 309. There is a function $\varepsilon(N)\to0$ such that

$$F(N)\ \ge\ \Bigl(\frac12-\varepsilon(N)\Bigr)\log N .$$

Hence $F(N)$ is not $o(\log N)$: the second question is answered no. With the trivial $F(N)<\log N+1$ the bound fixes the order of $F(N)$ only up to a factor between $\frac12$ and $1$; the asymptotic $F(N)\sim\log N$ is Yokota's 1997 theorem. The statement is recorded as Yokota's 1997 paper reports it (printed p. 162, in the paper's notation $|N(n)|$, which counts the empty sum $0$ and so exceeds $F(n)$ by one, without effect on the asymptotic bound): its introduction recalls that Erdős and Graham asked about the lower bound of $|N(n)|$ and says "This question was settled by the author [5]", with $|N(n)|\ge(\frac12-\varepsilon(n))\log n$ and $\varepsilon(n)\to0$, where [5] is the 1990 paper; the publisher's abstract of the 1990 paper ends by saying that it answers a question of Erdős and Graham. The 1990 paper is not held, so its own numbering of the count bound is not recorded; its Theorem 1, as quoted in the 1997 paper's Lemma 4 (p. 164), bounds the largest prime of the divisor set used to represent an integer $a$ by $\exp((a-1)/(1-1/\log\sigma_t-3/\log^2\sigma_t))$, and the 1997 proof of the asymptotic depends on it.

Covers. The second question: $F(N)$ is not $o(\log N)$. Not covered: the first question, which the problem page reads as asking for the asymptotic size of $F(N)$. This bound places $F(N)$ only between $(\frac12-o(1))\log N$ and $\log N+1$; the asymptotic is Yokota's 1997 theorem.

Acceptance. Refereed: Yokota, H., On number of integers representable as sums of unit fractions, Canad. Math. Bull. 33 (1990), no. 2, 235--241. The publisher's record dates the issue 1 June 1990, the day in this page's name. The site's commentary does not cite the paper, so no curator credit is listed; the first bound it credits is the 1997 theorem. The paper is not held, and no independent review of its proof is recorded in this corpus. The same conclusion follows from the three later results, each on its own claim page: Yokota's 1997 theorem, Croot's Main Theorem and Yokota's 2002 Corollary 1.