Open-problem statements, with their sources attached.

Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?

Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$

Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.

They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.

Is it true that for every $k$ there exists $n$ such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$

Brocard's Problem* Does $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$?

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).