Indeed, we must trivially have $\sum_{d|n_k}1/d \geq k$, or else there is a greedy colouring as a counterexample. Since $\prod_{p}(1+1/p^2)$ is finite we must have $\prod_{p|n_k}(1+1/p)\gg k$. To achieve the minimal $\prod_{p|n_k}p$ we take the product of primes up to $T$ where $\prod_{p\leq T}(1+1/p)\gg k$; by Mertens theorems this implies $T\geq C^{k}$ for some constant $C>1$, and hence $n_k\geq \prod_{p\mid n_k}p\geq \exp(cC^k)$ for some $c>0$.
Is it true that, for almost all $x$, for sufficiently large $n$, we have \[R_{n+1}(x)=R_n(x)+\frac{1}{m},\] where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)
Without the 'eventually' condition this can fail for some rational $x$ (although Erdős [Er50b] showed it holds without the eventually for rationals of the form $1/m$). For example \[R_1(\tfrac{11}{24})=\frac{1}{3}\] but \[R_2(\tfrac{11}{24})=\frac{1}{4}+\frac{1}{5}.\]
Kovač [Ko24b] has proved that this is false - in fact as false as possible: the set of $x\in (0,\infty)$ for which the best underapproximations are eventually 'greedy' has Lebesgue measure zero. (It remains an open problem to give any explicit example of a number which is not eventually greedy, despite the fact that almost all numbers have this property.)
Schinzel conjectured the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x<y<z$ such that \[\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]
Does this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?
Graham [Gr64b] has shown that $\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\equiv a\pmod{d}$ if and only if \[\left(\frac{n}{(n,(a,d))},\frac{d}{(a,d)}\right)=1.\] Does the greedy algorithm always terminate in such cases?
Graham [Gr64c] has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always.
Alekseyev [Al19] has proved this when $p(x)=x^2$, for all $m>8542$. For example, \[1=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{12}\] and \[200 = 2^2+4^2+6^2+12^2.\]
This conjecture would follow for all but at most finitely many exceptions if it were known that, for all large $N$, there exists a prime $p\in [N,2N]$ such that $\frac{p+1}{2}$ is also prime.
The smallest $b$ for each $a$ are listed at A375081 at the OEIS.
This was resolved in the affirmative by van Doorn [vD24], who proved $b=b(a)$ always exists, and in fact $b(a) \ll a$. Indeed, if $a\in (3^k,3^{k+1}]$ then one can take $b=2\cdot 3^{k+1}-1$. van Doorn also proves that $b(a)>a+(1/2-o(1))\log a$, and considers various generalisations of the original problem.
It seems likely that $b(a)\leq (1+o(1))a$, and perhaps even $b(a)\leq a+(\log a)^{O(1)}$.
More generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime $p\leq n$ divides $(a_n,L_n)$ if and only if $p$ divides the numerator of $1+\cdots+\frac{1}{k}$, where $k$ is the leading digit of $n$ in base $p$. This can be seen by writing \[a_n = \frac{L_n}{1}+\cdots+\frac{L_n}{n}\] and observing that the right-hand side is congruent to $1+\cdots+1/k$ modulo $p$. (The previous claim about $p-1$ follows immediately from Wolstenholme's theorem.)
This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of $\asymp\frac{x}{\log x}$ for the number of $n\in [1,x]$ such that $(a_n,L_n)=1$. In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.
The answer is yes, as proved by Martin [Ma00], who in fact proved that if $B=\mathbb{N}\backslash A$ then, for all large $x$, \[\frac{\lvert B\cap [1,x]\rvert}{x}\asymp \frac{\log\log x}{\log x},\] and also gave an essentially complete description of $B$ as those integers which are small multiples of prime powers.
van Doorn has observed that if $n\in A$ (with $n>1$) then $2n\in A$ also, since if $\sum \frac{1}{m_i}=1$ then $\frac{1}{2}+\sum\frac{1}{2m_i}=1$ also.
An elementary inductive argument shows that $n_k\leq ku_k$ where $u_1=1$ and $u_{i+1}=u_i(u_i+1)$, and hence \[v(k) \leq kc_0^{2^k},\] where \[c_0=\lim_n u_n^{1/2^n}=1.26408\cdots\] is the 'Vardi constant'.
Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl21] (coupled with the trivial greedy approach) implies that $k(N)=(1-o(1))\log N$.
Liu and Sawhney [LiSa24] have proved that $A(N)=(1-1/e+o(1))N$.
Estimate $f(N)$. In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$?
Wouter van Doorn has given an elementary argument that proves \[f(N)\leq (25/28+o(1))N.\] Indeed, consider the sets $S_a=\{2a,3a,4a,6a,12a\}\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\leq N/12$ and at least one element of $S_a$ when $N/12<a\leq N/6$, and an elementary calculation concludes the proof.
Stijn Cambie and Wouter van Doorn have proved that, if we allow solutions to this equation with non-distinct $b_i$, then the size of the maximal set is at most $N/2$. Indeed, this is the classical threshold for the existence of some distinct $a,b\in A$ such that $a\mid b$.
Estimate $f(N)$. In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$?
Wouter van Doorn has proved, in an unpublished note, that \[f(N) \leq (9/10+o(1))N.\] Stijn Cambie has observed that \[f(N)\geq (5/8+o(1))N,\] taking $A$ to be all odd integers $\leq N/4$ and all integers in $[N/2,N]$.
Stijn Cambie has also observed that, if we allow $b=c$, then there is a solution to this equation when $\lvert A\rvert \geq (\tfrac{2}{3}+o(1))N$, since then there must exist some $n,2n\in A$.
Related to [18].
This was solved by Yokota [Yo88], who proved that \[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\] This was improved by Liu and Sawhney [LiSa24] to \[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]
In [ErGr80] this problem is stated with the sequence $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, but Quanyu Tang has pointed out this is probably an error (since with that choice we do not have $\sum \frac{1}{u_n}=1$). This question with Sylvester's sequence is the most natural interpretation of what they meant to ask.
This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of $120$ form a counterexample. More generally, Sándor shows that for any $n\geq 2$ there exists a finite set $A\subseteq \mathbb{N}\backslash\{1\}$ with $\sum_{k\in A}\frac{1}{k}<n$ and no partition into $n$ parts each of which has $\sum_{k\in A_i}\frac{1}{k}<1$.
The minimal counterexample is $\{2,3,4,5,6,7,10,11,13,14,15\}$, found by Tom Stobart.
See also [321].
See also [320].
What if $a,b\in A$ with $a\neq b$ implies $a+b\nmid 2ab$? Must $\lvert A\rvert=o(N)$?
Wouter van Doorn has given an elementary argument that proves that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq (25/28+o(1))N$ then $A$ must contain $a\neq b$ with $a+b\mid ab$ (see the discussion in [301]).
See also [302].