Open-problem statements, with their sources attached.

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?

Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$.

More generally, for which pairs $x$ and $A$ does this process terminate?

Is every odd $n > 1$ the sum of a squarefree number and a power of 2?

Graham 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?

Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4.

Graham 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.

Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?

Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$

Is it true that $\lim_{N \to \infty} k(N) - (e - 1)N = \infty$?

1. There is NO good sequence with polynomial growth.

There are no examples known of the weakened coprime version if we insist that $1\not\in P\cup Q$.

1. There is a good sequence with sub-exponential growth.

Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with $\sum_{n \in A} 1/n > K$ there exists some $S \subseteq A$ such that $1 - \exp(-(c*K)) < \sum_{n \in S} 1/n \le 1$?

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.

What is the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$.

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$. What is $\Theta(c(N))$?

Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N)).

For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?