Open-problem statements, with their sources attached.

For what functions $g(N) → \infty$ is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant $c_k$?

Is it true that there is a constant $c_k$ such that for almost all $n < x$ we have $g_k(n)=c_k\log x+o(\log x)$?

Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?

If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.

How many iterations of $n\mapsto\phi(n) + 1$ are needed before a prime is reached?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. What is $\Theta(c(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

Can infinitely many $n$ reach the same prime under the iteration $n\mapsto\phi(n) + 1$?

What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto\phi(n) + 1$?

How many iterations of $n\mapsto\sigma(n) - 1$ are needed before a prime is reached?

If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ necessarily reaches a prime. Note: this is open , it is not clear that the σ iteration always terminates, since it is non-decreasing (unlike the φ iteration which is strictly decreasing).

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. What is $\Theta(c(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?

Is it true that iterates of $n\mapsto\sigma(n) - 1$ always reach a prime?

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 $σ_i(m) = σ_j(n)$?