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Claim. Let $F(N)$ be the count of Problem 309, the number of positive integers that are sums of distinct unit fractions with denominators at most $N$. For all large $N$,

$$F(N)\ \ge\ \Bigl\lfloor H_N-\Bigl(\frac{\pi^2}3+o(1)\Bigr)\frac{(\log\log N)^2}{\log N}\Bigr\rfloor,\qquad H_N=\sum_{n\le N}\frac1n.$$

The integer part is needed: every positive integer up to the bound inside it is representable, by the deduction on the library's corollary page from the paper's bound on the least $N$ at which a given integer becomes representable. The right side is at least $\log N+\gamma-1-o(1)$, so the count is not $o(\log N)$ and the second question is answered no. With Croot's upper floor $F(N)\le\lfloor H_N-(\frac12+o(1))(\log\log N)^2/\log N\rfloor$ (quoted in the same corollary from Croot's paper, not reproved) and his initial-segment description (his p. 2), for large $N$ the count is $m_N-1$ or $m_N$, where $m_N=\lfloor H_N\rfloor$. It is $m_N$ when the fractional part of $H_N$ exceeds $(\frac{\pi^2}3+o(1))(\log\log N)^2/\log N$, $m_N-1$ when it is below $(\frac12+o(1))(\log\log N)^2/\log N$, and undetermined in between. The lower bound is the first display of Corollary 1 on the library's result page, where the paper states it for $|N(n)|$, whose set includes the empty sum $0$, so that $|N(n)|=F(n)+1$; printed without the integer part, the bound holds for that count and fails for $F(n)$ infinitely often, whenever the fractional part of $H_N$ exceeds $(\frac{\pi^2}3+o(1))(\log\log N)^2/\log N$. The paper deduces it from its Theorem 1, the same bound for the representations whose denominators lie in a prescribed divisor set $D(t)$, through $|N^*(n)|\le|N(n)|$. The corollary's second display, an upper bound on the least $N$ at which a given integer becomes representable, bears on Problem 308 and is not part of this claim.

Depends on. the deduction on the library's Corollary 1 page.

Route. Given a large integer $a$, choose the stage $t$ at which the reciprocal sum over the divisor set $D(t)$ first exceeds $a$ by a prescribed small amount; the paper's Lemmas 4 and 5 bound the reciprocal mass outside $D(t)$ and the growth of the sum from one stage to the next, which places $a$ within $(\frac{\pi^2}3+o(1))(\log\log p)^2/\log p$ of $\log L(t)+\gamma$ for the stage's largest prime $p$ and cutoff $L(t)$; Lemma 3 writes the deficit as a sum of distinct divisors of the stage's product, whose cofactors are the denominators removed from the full sum over $D(t)$ to leave exactly $a$. Hence $a$ is representable with denominators at most $L(t)$, and the paper concludes (p. 357) its count bound for the representations with denominators in $D(t)$; that every integer up to $H_N-(\frac{\pi^2}3+o(1))(\log\log N)^2/\log N$ is representable with denominators at most $N$ is the deduction on the library's corollary page from the paper's bound on $F(a)$.

Acceptance. Refereed: Yokota, H., On the number of integers representable as sums of unit fractions, III, J. Number Theory 96 (2002), no. 2, 351--372. The publisher's record dates the issue October 2002 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 bound in the problem's commentary as the best lower bound known, credited to this paper; Bloom is independent of the author. Theorem 1 and Corollary 1 are recorded at statement depth, the proof in outline only, and no independent review of it is recorded in this corpus. The disproof is also carried by Yokota's 1997 theorem and by Croot's Main Theorem.