Open-problem statements, with their sources attached.

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $g = O(h)$ ?

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $h = O(g)$ ?

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $h = \Theta(g)$ ?

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $g = o(h)$ ?

Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct. How does $h(n)$ grow? Can we find a (good) explicit function $g$ such that $h = o(g)$ ?

Let $a_1< a_2 < ⋯$ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \to \infty$.

Let $a_1< a_2 < ⋯$ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \to 0$ for any $c > 0$.

Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j. Then it is conjectured that $a_k ~ \frac{k \log k}{\log \log k}$.

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?

Let $c > 0$ and $n$ be some large integer. What is the size of the largest set $A \subseteq \{1, \ldots, \lfloor c n \rfloor\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?

There is no consecutive triple of powerful numbers.

Erdős [Er76d] conjectured a stronger statement: if $n_k$ is the $k$th powerful number, then $n_{k+2} - n_k > n_k^c$ for some constant $c > 0$.

[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.

Are there any $2$-full $n$ such that $n+1$ is $3$-full?

Are there infinitely many 3-full $n$ such that $n+1$ is 2-full?

Are there any consecutive pairs of $3$-full integers?

Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n+1) > P(n)$ has density $\frac{1}{2}$.

Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.

Hickerson conjectured the largest solution the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, is 16!=14!5!2!.

Surányi was the first to conjecture that the only non-trivial solution to a!b!=n! is 6!7!=10!.