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Statement

Let $\mathcal L_1(r)$ be the set of integers $x>r^{-1}$ that cannot be the largest denominator $x_1$ in any Egyptian fraction representation $r=1/x_1+\cdots+1/x_t$ with $x_1>\cdots>x_t\ge1$ (p. 3), and let $L_1(r;x)=\#\{1\le n\le x:n\in\mathcal L_1(r)\}$.

Theorem 4 (p. 3). "Let $r$ be a positive rational number. The set $\mathcal L_1(r)$ has zero density, and in fact, if $x\ge3$ is a real number then

$$\frac{x\log\log x}{\log x}\ll_r L_1(r;x)\ll_r\frac{x\log\log x}{\log x}.$$

"

The specialization to Problem 292: with $A$ the set of $n$ for which $1=1/m_1+\cdots+1/m_k$ has a solution $1\le m_1<\cdots<m_k=n$, an integer $n\ge2$ lies in $A$ exactly when $n\notin\mathcal L_1(1)$ (a representation of $1$ with largest denominator $n\ge2$ cannot use the denominator $1$), and $1\in A$ by the one-term representation. So $B=\mathbb N\setminus A$ equals $\mathcal L_1(1)$, $|B\cap[1,x]|\asymp x\log\log x/\log x$, and $A$ has density $1$. The lower bound holds because tiny multiples of prime powers lie in $\mathcal L_1(r)$; the upper bound reflects the paper's finding that "all elements of $\mathcal L_1(r)$ are of this form (the only ambiguity being the exact meaning of 'tiny')" (p. 4), which is the site's "essentially complete description of $B$".

After the proof (p. 25) the paper records that the argument gives the explicit constants

$$\frac{(1+o_r(1))\,x\log\log x}{\log x}\le L_1(r;x)\le \frac{(24+o_r(1))\,x\log\log x}{\log x},$$

says that with much more care $24$ could be improved to $3$ but no further at present. It speculates, without proof, that for fixed $r$ and $\varepsilon>0$ only finitely many $n\in\mathcal L_1(r)$, written as $n=p^\nu m$ with $P^*(n)=p^\nu$, have $m\ge\log^{1+\varepsilon}p$, and notes that this would give $L_1(r;x)\sim x\log\log x/\log x$.

Source. G. Martin, Denser Egyptian fractions, arXiv:math/9811112v1 (18 November 1998), Theorem 4 on p. 3, read on the page image and in the text layer of that preprint; the proof is Section 7 (pp. 24--25). The journal version, Acta Arith. 95 (2000), no. 3, 231--260 (DOI 10.4064/aa-95-3-231-260), was not compared.

Read depth. Claims checked: the statement, the definition of $\mathcal L_1(r)$ and the counting function were read clause by clause on the page image of p. 3, the remark on p. 4 in the text layer, and the remark on p. 25 on the page image. The proof (pp. 24--25) was read for structure in the text layer; Lemmas 9, 10 and 18, which it invokes, were not checked.

Proof pointer

Lower bound (pp. 24--25): set $y=Cx/\log x$ with $C$ large. By Lemma 9, if $n\le x$ is the largest denominator of a representation of $r$ then $n$ has no prime factor larger than $y$ (once $x$ is so large that the primes dividing the denominator of $r$ are below $y$); so $\mathcal L_1(r)$ contains every $n\le x$ with $P(n)>y$, and the number of such $n$ is asymptotic to $x\log\log x/\log x$ by Lemma 10. Upper bound (p. 25): put $x'=x/\log x$ and $y=x'\log^{-23}x'$, and let $k$ be an integer with $x'<k\le x$ and $P^*(k)<y$. Then $r'=r-1/k$ lies in $I=[r/2,r]$ and its denominator satisfies $P^*(b')<y$, so Lemma 18 (the restatement of Proposition 5 in Section 6) gives a set $E\subseteq[1,x']$ with $\sum_{n\in E}1/n=r'$, and $E\cup\{k\}$ represents $r$ with largest denominator $k$. Hence for large $x$ every element of $\mathcal L_1(r)$ below $x$ lies in $\{n\le x'\}\cup\{n\le x:P^*(n)>x\log^{-24}x\}$, a set of size $\ll x\log\log x/\log x$.

Dependencies

Same-paper Lemma 9 (largest prime factor of a largest denominator), Lemma 10 (count of integers with a large prime factor) and Lemma 18 (Proposition 5, which rests on Propositions 7 and 8 of Sections 4 and 5).

Bears on

• Problem 292: with $r=1$ it shows that $A$ has density $1$ and that the exceptional set $B$ has counting function of exact order $x\log\log x/\log x$.
• Problem 285: context only; the two problems share this source.