Open-problem statements, with their sources attached.

Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}$.

Is it true that $(a_n,L_n)=1$ occurs for infinitely many $n$?

This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of $\asymp\frac{x}{\log x}$ for the number of $n\in [1,x]$ such that $(a_n,L_n)=1$.

In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.

If $A \subset \mathbb{N} has$\sum_{n \in A}\frac 1 n = \infty$, then must$A$ contain arbitrarily long arithmetic progressions?

Is it true that, for every $\varepsilon > 0$, $h(N) = \sqrt N + O_{\varespilon}(N^\varespilon)

Is it true that $N(b) \ll \log \log b$?

Let $\frac a b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \dots < n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$?

Are there two finite set of primes $P$ and $Q$ such that

?

Asked by Barbeau [Ba76].

[Ba76] Barbeau, E. J., Computer challenge corner: Problem 477: A brute force program.

Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: $\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}$?

It is conjectured that the set of primary pseudoperfect numbers is infinite.

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$

Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, $\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$ whenever the left-hand side is not zero?

Does there exist a set $A \subseteq \mathbb{N}$ such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ and every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \in A$?

Can the bound $O(\log N)$ be achieved for an additive complement to the primes? [Guy04] writes that Erdős offered $50 for the solution.

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $R(N)$?

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $\Theta(R(N))$?

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. Find the simplest $g(N)$ such that $R(N) = O(g(N))$.

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. Find the simplest $g(N)$ such that $R(N) = o(g(N))$.

Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$?

This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.

Is it true that if $m < k$ then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large $x$?