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We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice. Are there infinitely many practical $m$ such that $h(m) < (\log\log m)^{O(1)}?$ Is it true that $h(n!)<n^{o(1)}$?
It is easy to see that almost all numbers are not practical. This may be true with $m=n!$. Erdős originally showed that $h(n!) <n$. Vose [Vo85] has shown that $h(n!)\ll n^{1/2}$.

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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 $a\in A$.
$250 The density of integers which have two divisors$d_1,d_2$such that$d_1<d_2<2d_1$exists and is equal to$1$. The answer is yes (in fact with$2$replaced with any constant$c>1$), proved by Maier and Tenenbaum [MaTe84]. See also . 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$\sum_{n\leq x}\tau^*(n)$. Let$r(n)$count the number of$d_1,d_2$such that$d_1\mid n$and$d_2\mid n$and$d_1<d_2<2d_1$. Is it true that, for every$\epsilon>0$, $r(n) < \epsilon \tau(n)$ for almost all$n$, where$\tau(n)$is the number of divisors of$n$? See also . How large must$y=y(\epsilon,n)$be such that the number of integers in$(x,x+y)$with a divisor in$(n,2n)$is at most$\epsilon y$? For any$n$let$D_n$be the set of sums of the shape$d_1,d_1+d_2,d_1+d_2+d_3,\ldots$where$1<d_1<d_2<\cdots$are the divisors of$n$. 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\$.