Open-problem statements, with their sources attached.

Let A ⊆ ℕ be an infinite set such that the triple sums a + b + c are all distinct for a, b, c in A (aside from the trivial coincidences). Is it true that liminf n → ∞ |A ∩ {1, …, N}| / N^(1/3) = 0?

Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$.

Is it true that $\lim_{k → ∞} σ_k(n)^{\frac 1 k} = ∞$?

This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory", 1990).

Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$. Is it true that, for every $m, n ≥ 2$, there exist some $i, j$ such that $σ_i(m) = σ_j(n)$?

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].