Logo
All Random Solved Random Open
18 solved out of 48 shown (show only solved or open)
SOLVED
Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?
This follows from the colouring result of Croot [Cr03]. Obtaining better quantitative aspects would be interesting - note that Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$. Is there a better bound?
SOLVED
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
The answer is yes, as proved by Croot [Cr03].
SOLVED - $100
If $\delta>0$ and $N$ is sufficiently large in terms of $\delta$, and $A\subseteq\{1,\ldots,N\}$ is such that $\sum_{a\in A}\frac{1}{a}>\delta \log N$ then must there exist $S\subseteq A$ such that $\sum_{n\in S}\frac{1}{n}=1$?
Solved by Bloom [Bl21], who showed that the quantitative threshold \[\sum_{n\in A}\frac{1}{n}\gg \frac{\log\log\log N}{\log\log N}\log N\] is sufficient. This was improved by Liu and Sawhney [LiSa24] to \[\sum_{n\in A}\frac{1}{n}\gg (\log N)^{4/5+o(1)}.\] Erdős speculated that perhaps even $\gg (\log\log N)^2$ might be sufficient. (A construction of Pomerance, as discussed in the appendix of [Bl21], shows that this would be best possible.)
OPEN
Let $F(k)$ be the number of solutions to \[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\] where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.
The current best bounds known are \[2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k},\] where $c>0$ is some absolute constant and $c_0=1.26408\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin [Ko14] and the upper bound to Elsholtz and Planitzer [ElPl21].
OPEN
Let $x>0$ be a real number. For any $n\geq 1$ let \[R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x\] be the maximal sum of $n$ distinct unit fractions which is $<x$.

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?)

Erdős and Graham write it is 'not difficult' to construct irrational $x$ such that this fails. Curtiss [Cu22] showed that this is true for $x=1$ and Erdős [Er50b] showed it is true for all $x\in\mathbb{Q}$.

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 \[R_2(\tfrac{11}{24})=\frac{1}{4}+\frac{1}{5}.\]

OPEN
For every $n\geq 2$ there exist distinct integers $1\leq x<y<z$ such that \[\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]
The Erdős-Straus conjecture. The existence of a representation of $4/n$ as the sum of at most four distinct unit fractions follows trivially from a greedy algorithm.

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 \[\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]

OPEN
Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ such that $n\geq 1/x$ and repeat with $x$ replaced by $x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$.

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?

In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\mathbb{N}$. The problem when $A$ is the set of odd numbers is due to Stein.

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.

Additional thanks to: Zachary Hunter
OPEN
Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for all $n\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\leq n_1<\cdots <n_k$ such that \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] and \[m=p(n_1)+\cdots+p(n_k)?\]
Graham [Gr63] has proved this when $p(x)=x$. Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$. Burr has proved this if $p(x)=x^k$ with $k\geq 1$ and if we allow $n_i=n_j$.
SOLVED
Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<\cdots <n_k$ with \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Is it true that \[f(k)=(1+o(1))\frac{k}{e-1}?\]
The upper bound $f(k) \leq (1+o(1))\frac{k}{e-1}$ is trivial since for any $u\geq 1$ we have \[\sum_{u\leq n\leq eu}\frac{1}{n}=1+o(1),\] and hence if $f(k)=u$ then we must have $k\geq (e-1-o(1))u$. Essentially solved by Croot [Cr01], who showed that for any $N>1$ there exists some $k\geq 1$ and \[N<n_1<\cdots <n_k \leq (e+o(1))N\] with $1=\sum \frac{1}{n_i}$.
Additional thanks to: Zachary Hunter
SOLVED
Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<\cdots <n_k$ with \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Is it true that \[f(k)=(1+o(1))\frac{e}{e-1}k?\]
It is trivial that $f(k)\geq (1+o(1))\frac{e}{e-1}k$, since for any $u\geq 1$ \[\sum_{e\leq n\leq eu}\frac{1}{n}= 1+o(1),\] and so if $eu\approx f(k)$ then $k\leq \frac{e-1}{e}f(k)$. Proved by Martin [Ma00].
Additional thanks to: Zachary Hunter
SOLVED
Let $k\geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and integers $n_1<\cdots<n_k\in I$ such that \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?\]
The answer is yes, proved by Croot [Cr01].
OPEN
Let $k\geq 2$. Is it true that, for any distinct integers $n_1<\cdots <n_k$ such that \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] we must have $\max(n_{i+1}-n_i)\geq 3$?
The example $1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$ shows that $3$ would be best possible here. The lower bound of $\geq 2$ is equivalent to saying that $1$ is not the sum of reciprocals of consecutive integers, proved by Erdős [Er32].

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.

OPEN
Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that \[\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?\]
This is still open even if $\lvert I_2\rvert=1$. It is perhaps true with two intervals replaced by any $k$ intervals.
OPEN
Is it true that, for all sufficiently large $k$, there exist intervals $I_1,\ldots,I_k$ with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that \[1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?\]
OPEN
Let $a\geq 1$. Must there exist some $b>a$ such that \[\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},\] with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?
For example, \[\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.\]
OPEN
Let $n\geq 1$ and define $L_n$ to be the lowest common multiple of $\{1,\ldots,n\}$ and $a_n$ by \[\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.\] Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?
OPEN
Let $A$ be the set of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$. Explore $A$. In particular,
  • Does $A$ have density $1$?
  • What are those $n\in A$ not divisible by any $d\in A$ with $1<d<n$?
Straus observed that $A$ is closed under multiplication. Furthermore, it is easy to see that $A$ does not contain any prime power.
Additional thanks to: Zachary Hunter
OPEN
Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$.
Results of Bleicher and Erdős [BlEr75] imply $v(k) \gg k!$. It may be that $v(k)$ grows doubly exponentially in $\sqrt{k}$ or even $k$.

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'.

Additional thanks to: Zachary Hunter
SOLVED
Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] with $t=n_1<\cdots <n_k\leq N$. Estimate $t(N)$.
Erdős and Graham [ErGr80] could show \[t(N)\ll\frac{N}{\log N},\] but had no idea of the true value of $t(N)$.

Solved by Liu and Sawhney [LiSa24] (up to $(\log\log N)^{O(1)}$), who proved that \[\frac{N}{(\log N)(\log\log N)^3(\log\log\log N)^{O(1)}}\ll t(N) \ll \frac{N}{\log N}.\]

OPEN
Let $N\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\leq n_1<\cdots <n_k$ with \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Is it true that \[\lim_{N\to \infty} k(N)-(e-1)N=\infty?\]
Erdős and Straus [ErSt71b] have proved the existence of some constant $c>0$ such that \[-c < k(N)-(e-1)N \ll \frac{N}{\log N}.\]
SOLVED
Let $N\geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,\ldots,A_k\subseteq \{1,\ldots,N\}$ with $\sum_{n\in A_i}\frac{1}{n}=1$ for all $i$. Estimate $k(N)$. Is it true that $k(N)=o(\log N)$?
More generally, how many disjoint $A_i$ can be found in $\{1,\ldots,N\}$ such that the sums $\sum_{n\in A_i}\frac{1}{n}$ are all equal? Using sunflowers it can be shown that there are at least $N\exp(-O(\sqrt{\log N}))$ such sets.

Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] (coupled with the trivial greedy approach) implies that $k(N)=(1-o(1))\log N$.

Additional thanks to: Zachary Hunter and Mehtaab Sawhney
SOLVED
Let $N\geq 1$. How many $A\subseteq \{1,\ldots,N\}$ are there such that $\sum_{n\in A}\frac{1}{n}=1$?
It was not even known for a long time whether this is $2^{cN}$ for some $c<1$ or $2^{(1+o(1))N}$. In fact the former is true, and the correct value of $c$ is now known.
  • Steinerberger [St24] proved the relevant count is at most $2^{0.93N}$;
  • Independently, Liu and Sawhney [LiSa24] gave both upper and lower bounds, proving that the count is \[2^{(0.91\cdots+o(1))N},\] where $0.91\cdots$ is an explicit number defined as the solution to a certain integral equation;
  • Again independently this same asymptotic was proved (with a different proof) by Conlon, Fox, He, Mubayi, Pham, Suk, and Verstraëte [CFHMPSV24], who prove more generally, for any $x\in \mathbb{Q}_{>0}$, a similar expression for the number of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in A}\frac{1}{n}=x$;
  • The above papers all appeared within weeks of each other in 2024; in 2017 a similar question (with $\leq 1$ rather than $=1$) was asked on MathOverflow by Mikhail Tikhomirov and proofs of the correct asymptotic were sketched by Lucia, RaphaelB4, and js21.
Additional thanks to: Zachary Chase
SOLVED
Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$?
The answer is yes, proved by Bloom [Bl21].
SOLVED
Is there an infinite sequence $a_1<a_2<\cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to $1$?
There does not exist such a sequence, which follows from the positive solution to [298] by Bloom [Bl21].
SOLVED
Let $A(N)$ denote the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$. Estimate $A(N)$.
Erdős and Graham believe the answer is $A(N)=(1+o(1))N$. Croot [Cr03] disproved this, showing the existence of some constant $c<1$ such that $A(N)<cN$ for all large $N$. It is trivial that $A(N)\geq (1-\frac{1}{e}+o(1))N$.

Liu and Sawhney [LiSa24] have proved that $A(N)=(1-1/e+o(1))N$.

Additional thanks to: Zachary Hunter
OPEN
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that for any $a,b_1,\ldots,b_k\in A$ with $k\geq 2$ we have \[\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}?\] Is there a constant $c>1/2$ such that $\lvert A\rvert >cN$ is possible for all large $N$?
The example $A=(N/2,N]\cap \mathbb{N}$ shows that $\lvert A\rvert\geq N/2$.
Additional thanks to: Zachary Hunter
OPEN
Is it true that, for any $\delta>1/2$, if $N$ is sufficiently large and $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert \geq \delta N$ then there exist $a,b,c\in A$ such that \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}.\]
The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].

The possible alternative question, that if $A\subseteq \mathbb{N}$ is a set of positive lower density then must there exist $a,b,c\in A$ such that \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c},\] has a negative answer, taking for example $A$ to be the union of $[5^k,(1+\epsilon)5^k]$ for large $k$ and sufficiently small $\epsilon>0$. This was observed by Hunter and Sawhney.

Additional thanks to: Zachary Hunter and Mehtaab Sawhney
SOLVED
Is it true that in any finite colouring of the integers there exists a monochromatic solution to \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}\] with distinct $a,b,c$?
The density version of this is [302]. This colouring version is true, as proved by Brown and Rödl [BrRo91].
OPEN
For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with \[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Estimate $N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that $N(b) \ll \log\log b$?
Erdős [Er50c] proved that \[\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.\] The upper bound was improved by Vose [Vo85] to \[N(b) \ll \sqrt{\log b}.\] One can also investigate the average of $N(a,b)$ for fixed $b$, and it is known that \[\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.\]

Related to [18].

Additional thanks to: Zachary Hunter
SOLVED
For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with \[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\] Estimate $D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that \[D(b) \ll b(\log b)^{1+o(1)}?\]
Bleicher and Erdős [BlEr76] have shown that \[D(b)\ll b(\log b)^2.\] If $b=p$ is a prime then \[D(p) \gg p\log p.\]

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 \[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]

OPEN
Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that \[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?\]
For $n_i$ the product of three distinct primes, this is true when $b=1$, as proved by Butler, Erdős and Graham [BuErGr15] (this paper is perhaps Erdős' last paper, appearing 19 years after his death).
OPEN
Are there two finite sets of primes $P,Q$ such that \[1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?\]
Asked by Barbeau [Ba76]. Can this be done if we drop the requirement that all $p\in P$ are prime and just ask for them to be relatively coprime, and similarly for $Q$?
OPEN
Let $N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Is it true that the set of integers not representable as such has the shape $[m,\infty)$ for some $m$?
SOLVED
Let $N\geq 1$. How many integers can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Are there $o(\log N)$ such integers?
The answer to the second question is no: there are at least $(1-o(1))\log N$ many such integers, which follows from a more precise result of Croot [Cr99], who showed that every integer at most \[\leq \sum_{n\leq N}\frac{1}{n}-(\tfrac{9}{2}+o(1))\frac{(\log\log N)^2}{\log N}\] can be so represented.
Additional thanks to: Zachary Hunter and Mehtaab Sawhney
SOLVED
Let $\alpha >0$ and $N\geq 1$. Is it true that for any $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert \geq \alpha N$ there exists some $S\subseteq A$ such that \[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\] with $a\leq b =O_\alpha(1)$?
Liu and Sawhney [LiSa24] observed that the main result of Bloom [Bl21] implies a positive solution to this conjecture. They prove a more precise version, that if $(\log N)^{-1/7+o(1)}\leq \alpha \leq 1/2$ then there is some $S\subseteq A$ such that \[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\] with $a\leq b \leq \exp(O(1/\alpha))$. They also observe that the dependence $b\leq \exp(O(1/\alpha))$ is sharp.
OPEN
What is the minimal value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no $S$ such that $\sum_{n\in S}\frac{1}{n}=1$? Is it \[e^{-(c+o(1))N}\] for some constant $c\in (0,1)$?
It is trivially at least $1/[1,\ldots,N]$.
OPEN
Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of integers with $\sum_{n\in A}\frac{1}{n}>K$ there exists some $S\subseteq A$ such that \[1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?\]
Erdős and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.
Additional thanks to: Mehtaab Sawhney
OPEN
Are there infinitely many solutions to \[\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},\] where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are distinct primes?
For example, \[\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}.\]
SOLVED
Let $n\geq 1$ and let $m$ be minimal such that $\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$. We define \[\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1.\] How small can $\epsilon(n)$ be? Is it true that \[\liminf n^2\epsilon(n)=0?\]
Solved by Steinerberger [St24b] who proved that in fact, contrary to the expectations of Erdős and Graham, $\liminf n^2\epsilon(n)>0$, and explicitly proved that for all sufficiently large $n$ \[\epsilon(n)\geq \frac{1}{101n^2}.\]
OPEN
Let $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$. Let $a_1<a_2<\cdots $ be any other sequence with $\sum \frac{1}{a_k}=1$. Is it true that \[\liminf a_n^{1/2^n}<\lim u_n^{1/2^n}=c_0=1.264085\cdots?\]
$c_0$ is called the Vardi constant and the sequence $u_n$ is Sylvester's sequence.

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.

Additional thanks to: Quanyu Tang
SOLVED
Is it true that if $A\subset \mathbb{N}\backslash\{1\}$ is a finite set with $\sum_{n\in A}\frac{1}{n}<2$ then there is a partition $A=A_1\sqcup A_2$ such that \[\sum_{n\in A_i}\frac{1}{n}<1\] for $i=1,2$?
This is not true if $A$ is a multiset, for example $2,3,3,5,5,5,5$.

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.

Additional thanks to: Tom Stobart
OPEN
Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with \[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\] Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, \[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}\] whenever the left-hand side is not zero?
Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example \[\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.\]
Additional thanks to: Zachary Chase
OPEN
Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite $S\subset A$ such that \[\sum_{n\in S}\frac{f(n)}{n}=0?\] What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?
Erdős and Straus [ErSt75] proved this when $A=\mathbb{N}$. Sattler [Sat75] proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since \[\sum_{k\geq 2}\frac{1}{k^2}<1.\]
OPEN
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that \[\sum_{n\in A}\frac{\delta_n}{n}=0\] and \[\sum_{n\in A'}\frac{\delta_n}{n}\neq 0\] for all $A'\subsetneq A$?
OPEN
Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$.
Bleicher and Erdős [BlEr75] proved the lower bound \[\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq \frac{\log S(N)}{\log 2},\] valid for $k\geq 4$ and $\log_kN\geq k$, and also [BlEr76b] proved the upper bound \[\log S(N)\leq \log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),\] valid for $r\geq 1$ and $\log_{2r}N\geq 1$. (In these bounds $\log_in$ denotes the $i$-fold iterated logarithm.)

See also [321].

Additional thanks to: Boris Alexeev and Dustin Mixon
OPEN
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that all sums $\sum_{n\in S}\frac{1}{n}$ are distinct for $S\subseteq A$?
Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that \[\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),\] valid for any $k\geq 4$ with $\log_kN\geq k$ and any $r\geq 1$ with $\log_{2r}N\geq 1$. (In these bounds $\log_in$ denotes the $i$-fold iterated logarithm.)

See also [320].

Additional thanks to: Boris Alexeev, Zachary Hunter, and Dustin Mixon
OPEN
Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ then $a+b\nmid ab$. Can $A$ be 'substantially more' than the odd numbers?

What if $a,b\in A$ with $a\neq b$ implies $a+b\nmid 2ab$? Must $\lvert A\rvert=o(N)$?

The connection to unit fractions comes from the observation that $\frac{1}{a}+\frac{1}{b}$ is a unit fraction if and only if $a+b\mid ab$.
OPEN
Let $A\subseteq \mathbb{N}$ be a lacunary sequence (so that $A=\{a_1<a_2<\cdots\}$ and there exists some $\lambda>1$ such that $a_{n+1}/a_n\geq \lambda$ for all $n\geq 1$). Must \[\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}\] contain all rationals in some open interval?
Bleicher and Erdős conjecture the answer is no.