Open-problem statements, with their sources attached.

Are there infinitely many barriers for ω?

Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity.

Does there exist some ε > 0 such that there are infinitely many ε-barriers for ω?

Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m) = h_j(n)$?

Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Does V(2x)/V(x)→2 ?

Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Is there an asymptotic formula for V(x)?

Let$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$and$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$ Does $\lim V(x)/V'(x)$ exist?

Formalization note: We formalize the limit of the inverse fraction V'(x)/V(x) to ensure the limit is finite (bounded between 0 and 1).

Is it $>1$?

It is open whether the set of non-cototients has positive density.

Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?

Does $f(n)$ miss infinitely many integers?

Is $f$ surjective?

How does $f$ grow?

Does $f$ become stationary at some point?

Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$. Is it true that the set of integers which eventually appear has positive density?

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$

Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?

This is discussed in this MathOverflow question [MathOverflow].

Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?

Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?

Is it true that $m_n<p_n$ for almost all $n$?