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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. For all large $N$,

$$\Bigl(1-\frac{5\log\log N}{\log N}\Bigr)\log N\ \le\ F(N)\ <\ \log N+1,$$

that is, $F(N)\ge\log N-5\log\log N$, the upper bound being the trivial one. Hence $F(N)\sim\log N$: the count is not $o(\log N)$, the second question is answered no, and the first is answered to leading order. This is Theorem 1 of the paper, in the paper's notation $|N(n)|$, which counts the empty sum $0$ and so exceeds $F(n)$ by one without affecting the asymptotic bounds, and $\log_2$ for the iterated logarithm, on the library's result page. The proof opens by stating more (p. 167): for large $N$ every positive integer up to $\log N-5\log\log N$ is representable, so $N(N)$ contains that initial segment; Croot's 1999 paper states this range citing the 1998 Corrigendum and uses the paper with its Corrigendum for the integers below a fixed bound. The printed steps reach only the integers up to $\log N-(3+o(1))\log N/\log\log N$ (see Route). The theorem sharpens Yokota's 1990 bound $F(N)\ge(\frac12-o(1))\log N$, the first disproof, which the paper's introduction (p. 162) reports as having settled the question.

Depends on. Yokota's 1990 theorem: the paper's Lemma 4 (p. 164), the bound on the largest prime of the divisor set used to represent $a$, is quoted as Theorem 1 of the 1990 paper without proof.

Route. For a large integer $a$, denominators are drawn from the divisors of a product of prime powers $p^{2^i}$ and consecutive primes chosen so that the reciprocal sum over the divisors up to a cutoff falls just short of $a$; the deficit is a rational whose numerator is written, through the paper's Lemma 5, as a sum of distinct divisors of the product, which supplies the missing reciprocals with denominators at most $a^4\exp[a(1+3/\log a)]$. The paper then says that this being at most $N$ implies $a\le\log N(1-5\log\log N/\log N)$ (p. 168), the converse of the step the theorem needs; the bound is at most $N$ only for $a$ up to $\log N-(3+o(1))\log N/\log\log N$, enough for $F(N)\sim\log N$, so the answer no stands, but not for the stated error term $5\log\log N$ (the bound itself also follows from Croot's Main Theorem; see Acceptance).

Acceptance. Refereed: Yokota, H., On number of integers representable as a sum of unit fractions, II, J. Number Theory 67 (1997), no. 2, 162--169, with a Corrigendum, J. Number Theory 72 (1998), 150. The publisher's record dates the issue December 1997 and gives no day; the day in this page's name is the first of that month. Reviewed: the site's curator, Thomas Bloom, marks Problem 309 disproved and credits this paper's lower bound, $\log N-O(\log\log N)$, in the problem's commentary. Theorem 1 and the opening of its proof are recorded at statement depth, the rest of the proof in outline only; the Corrigendum is not held, so what it corrects is not recorded. No independent review of the proof is recorded in this corpus. The same conclusion follows from Croot's Main Theorem and from Yokota's 2002 Corollary 1, each on its own claim page.

Formalization. The file src/latest/ErdosProblems/Erdos309.lean in Boris Alexeev's lean-proofs collection at the pinned commit (the second link) declares itself a Lean formalization of a solution to Problem 309, names Yokota, Croot and Thomas Bloom as its informal authors and Codex, GPT-5.6 Sol (OpenAI Codex) as its formal authors. Its final theorem proves $F(N)/\log N\to1$ and that $F(N)$ is not $o(\log N)$, the conclusion of this claim (its $F$ counts the empty sum $0$, as the paper's $|N(n)|$ does), by a packing argument built on a unit-fraction extraction theorem rather than by this paper's construction; the formal-conjectures statement file for the problem points at it, and the Lean suffix of the site's label refers to it, as the problem page records. The corpus did not build the file, and only its top file is recorded, so no formalized evidence is listed; Bloom's place among the informal authors credits a theorem the development uses and is not a review of this paper. The same link is on Croot's page.