Open-problem statements, with their sources attached.

Is $n! - 1$ powerful for finitely many $n$?

Let $A=\{n_1 < n_2 < \cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?

If $r≥4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful?

If $r≥4$ are there infinitely many sums of $r-2$ coprime $r$-powerful numbers that are themselves $r$-powerful?

Are there infinitely many triples of coprime $3$-powerful numbers $a, b, c$ such that $a + b = c$?

Let $r \ge 3$. Is it true that the set of integers which are the sum of at most $r$ $r$-powerful numbers has density $0$?

Is it true that the set of integers which are the sum of at most three cubes has density $0$?

It is not known if all large integers are the sum of at most $r$-many $r$-powerful numbers.

Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$.

Let $A$ be the set of powerful numbers. Is is true that $1_A\ast 1_A(n)=n^{o(1)}$ for every $n$?

Let $k \ge 4$ and $r\ge 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?

Let $k \ge 4$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>1$?

This was conjectured by Dirac in 1970.

The case $k=4$ and $r=1$ remains open: Are there $4$-critical graphs without any critical edges?

Is it true that $F(x) \leq (\log x)^{O(1)}$?

Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors?

Is it true that $\liminf f(n)=1$?

Is it true that $\limsup f(n)=\infty$?

Is it true that $f(n)=o(\log\log n)$ for all $n$?

Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\epsilon>0$, if $x$ is sufficiently large there exists $y<x$ such that $\pi(x)< \pi(y)+\epsilon \pi(x-y)$. Compare this to [855].

The study of $f(p)$ is even harder, and Erdős could not prove that $\sum_{p<x}f(p)^2\sim \pi(x)$.