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Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$?

Asked by Erdős and Tenenbaum. Ruzsa has found a counterexample. Tenenbaum asked the weaker variant (still open) where for every $\epsilon>0$ there is some $k=k(\epsilon)$ such that at least $1-\epsilon$ density of all integers have a divisor of the form $a+k$ for some $a\in A$.

Let $A\subseteq\mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$. Is it true that, for every $k$,
\[\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k}=\infty?\]

The answer is yes, proved by Erdős and Sárkőzy [ErSa80].

Let $\delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. What is the growth rate of $\delta(n)$?

If $\delta'(n)$ is the density of integers which have exactly one divisor in $(n,2n)$ then is it true that $\delta'(n)=o(\delta(n))$?

Erdős [Er35] proved that $\delta(n)<(\log n)^{-c}$ for some $c>0$. Tenenbaum [Te75] showed that $\delta(n)=(\log n)^{(-1+o(1))\alpha}$ where
\[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\]

This has been resolved by Ford [Fo08]. Among many other results, Ford proves \[\delta(n)\asymp \frac{1}{(\log n)^\alpha(\log\log n)^{3/2}},\] and that the second conjecture is false (i.e. $\delta'(n) \geq \delta(n)$ for some constant $c>0$)

Let $\tau(n)$ count the divisors of $n$ and $\tau^*(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$. Is it true that, for all $\epsilon>0$,
\[\tau^*(n) < \epsilon \tau(n)\]
for almost all $n$?

Erdős and Graham also ask whether there is a good inequality known for $\sum_{n\leq x}\tau^*(n)$.

What is the size of $D_n\backslash \cup_{m<n}D_m$?

If $f(N)$ is the minimal $n$ such that $N\in D_n$ then is it true that $f(N)=o(N)$? Perhaps just for almost all $N$?

Let $A$ be the set of all $n$ such that $n=d_1+\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\mid n$. Does
\[\sum_{n\in A}\frac{1}{n}\]
converge?

The same question can be asked for those $n$ which do not have distinct sums of sets of divisors, but any proper divisor of $n$ does.

Call $n$ weird if $\sigma(n)\geq 2n$ and $n\neq d_1+\cdots+d_k$, where the $d_i$ are distinct proper divisors of $n$. Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?

Weird numbers were investigated by Benkoski and Erdős [BeEr74], who proved that the set of weird numbers has positive density.

Melfi [Me15] has proved that there are infinitely many primitive weird numbers, conditional on various well-known conjectures on the distribution of prime gaps. For example, it would suffice to show that $p_{n+1}-p_n <\frac{1}{10}p_n^{1/2}$ for sufficiently large $n$.