Open-problem statements, with their sources attached.

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$?

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity.

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded.

Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$?

Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$).

Is it true that $C(x)=x^{1-o(1)}$?

This is discussed in problem A13 of Guy's collection [Gu04].

Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$.

Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$?

The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower than any power of $n^(\frac{1}{\log\log n})$. The second bound is that $f(n)$ is at most a power of $\log n$.

Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$.

How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? Is it true that this number is asymptotic to c * x for some constant c > 0?

Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational.

Estimate $n_k$ by finding a better upper bound.

Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$?

This is mentioned as B46 in Unsolved Problems in Number Theory by Richard K. Guy

Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$?

Is it true that there are infinitely many $p$ for which $f(p) = p − 1$?

Is it true that $f(p)/p \to 0$ for $p \to \infty$ in a density 1 subset of the primes?

Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \le x$ for which $f(p)=p−1$ is $o(x/\log x)$.

[HaSu02] Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559.