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Claim. Let $n(N)$ be the largest integer such that every integer $1\le n\le n(N)$ is a sum of distinct unit fractions with denominators at most $N$, and $H_N=\sum_{n\le N}1/n$. Croot's Main Theorem, on the library's result page, gives for all large $N$

$$\Bigl\lfloor H_N-\tfrac92(1+o(1))\tfrac{(\log\log N)^2}{\log N}\Bigr\rfloor \le n(N)\le \Bigl\lfloor H_N-\tfrac12(1+o(1))\tfrac{(\log\log N)^2}{\log N}\Bigr\rfloor.$$

For the count $F(N)$ of Problem 309 the deduction is one line, written on this page and not in the paper: $F(N)\ge n(N)$ and $H_N=\log N+\gamma+O(1/N)$ give

$$F(N)\ \ge\ H_N-\tfrac92(1+o(1))\tfrac{(\log\log N)^2}{\log N}-1 \ =\ \log N+\gamma-1-o(1),$$

and with the trivial $F(N)\le H_N<\log N+1$ this is $F(N)=\log N+O(1)$, so $F(N)$ is not $o(\log N)$ and the second question is answered no. In fact the paper's p. 2 shows that the representable integers form an initial segment for large $N$, so $F(N)=n(N)$ and both floors bound $F(N)$ itself, which answers the first question to within the window between the two floors. Croot's conjecture that the upper floor is the truth would close it.

Depends on. Yokota's 1997 theorem supplies, in the proof of the lower bound, the representability of the integers below a fixed bound (the paper's references [5] and [6]).

Route. Lower bound: starting from the full harmonic sum, remove as few terms as possible to leave an integer, prime power by prime power from the top, with the removed reciprocal mass controlled by a quantity built from prime powers (Propositions 1 and 3); the integer reached changes by at most one when $N$ increases by one, so every integer between the value for a fixed $N_0$ and the value for $N$ is reached at some intermediate bound, and the integers below the fixed value are representable by Yokota's theorem. Upper bound: a $p$-adic argument shows that no denominator in an integer reciprocal sum has a prime factor above $N\log\log N/\log N$, and the reciprocal mass of the integers with such a factor exceeds $\frac12(1+o(1))(\log\log N)^2/\log N$.

Acceptance. Refereed: Croot, III, Ernest S., On some questions of Erdős and Graham about Egyptian fractions, Mathematika 46 (1999), no. 2, 359--372. The publisher's record dates the issue December 1999 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 records this theorem in the problem's commentary, crediting Croot with the representability of every integer up to $H_N-(\frac92+o(1))(\log\log N)^2/\log N$; Bloom is independent of the author. The locators are pages of the author's fourteen-page typescript, posted on the author's papers page (the second link); the journal text has not been compared, the proof is recorded in outline only, and no independent review of it is recorded in this corpus. The constant $\frac92$ of the lower floor is improved to $\frac{\pi^2}3$ by Yokota's 2002 Corollary 1.

Formalization. The file src/latest/ErdosProblems/Erdos309.lean in Boris Alexeev's lean-proofs collection at the pinned commit (the third link) declares itself a Lean formalization of a solution to Problem 309, names Croot, Yokota and Thomas Bloom as its informal authors and Codex, GPT-5.6 Sol (OpenAI Codex) as its formal authors. It proves the conclusion drawn on this page, $F(N)/\log N\to1$ and $F(N)$ not $o(\log N)$, by a packing argument built on a unit-fraction extraction theorem rather than by the Main Theorem's construction, and does not prove the Main Theorem's floors. The corpus did not build the file, and only its top file is recorded, so no formalized evidence is listed; the same link is on Yokota's 1997 page.