Open-problem statements, with their sources attached.

There are no examples known of the weakened coprime version if we insist that $1\not\in P\cup Q$.

1. There is a good sequence with sub-exponential growth.

Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with $\sum_{n \in A} 1/n > K$ there exists some $S \subseteq A$ such that $1 - \exp(-(c*K)) < \sum_{n \in S} 1/n \le 1$?

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.

Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?

For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?

Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?

Let $A \subset \mathbb{N}$ be an additive basis of order 2.

Must there exist $B = \{b_1 < b_2 < \dots\} \subseteq A$ which is also a basis such that $\lim_{k\to\infty} \frac{b_k}{k^2}$ does not exist?

Erdős Problem 1113.* Do there exist Sierpiński numbers that possess no finite covering set of primes?

Erdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply that there are infinitely many Fermat primes.

Erdős Problem 329.* Let A ⊆ ℕ be a Sidon set. How large can lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2} be?

Filaseta–Finch–Kozek conjecture (2008).* Every Sierpiński number is either a perfect power or possesses a finite covering set of primes.

The converse: if the maximum density is 1, then any finite Sidon set can be embedded in a perfect difference set modulo $n > 0$.

Since the consequent is false (due to the counterexamples in [Ha47] and [AlMi25]), this implication is logically equivalent to the statement that the maximum upper density of Sidon sets is NOT 1. Because the maximum upper density problem is still open, the truth value of this implication is also an open research problem.

The Collatz conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is 1.

Does there exist a minimal basis $A \subset \mathbb{N}$ with positive density such that, for any $n \in A$, the (upper) density of integers which cannot be represented without using $n$ is positive?

Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that $\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0$ as $x\to \infty$?

Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that $\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?$

Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$?

The only known such $n$ are $4, 7, 15, 21, 45, 75, 105$ (OEIS A039669).

Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?

In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form $2^m3^n$ and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."

Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \in A} \frac{1}{n} < \infty$?