Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that \[d_n<d_{n+1}<\cdots <d_{n+m}\] and infinitely many $n$ such that \[d_n> d_{n+1}>\cdots >d_{n+m}.\]
Are there infinitely many practical $m$ such that \[h(m) < (\log\log m)^{O(1)}?\] Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?
The sequence of practical numbers is A005153 in the OEIS.
Proved by Sárkzözy [Sa85] for all sufficiently large $n$, and by Granville and Ramaré [GrRa96] for all $n\geq 5$.
More generally, if $f(n)$ is the largest integer such that, for some prime $p$, we have $p^{f(n)}$ dividing $\binom{2n}{n}$, then $f(n)$ should tend to infinity with $n$. Can one even disprove that $f(n)\gg \log n$?
In [Er79] Erdős says perhaps $s_{n+1}-s_n \ll \log s_n$, but he is 'very doubtful'.
Filaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.
Schinzel conjectured the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x<y<z$ such that \[\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]
A positive answer would imply that \[\sum_{p\leq n}1_{p\mid \binom{2n}{n}}\frac{1}{p}=(1-o(1))\log\log n,\] and Erdős, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true.
Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \omega(m)\leq n$ for all $m<n$?
Erdős also believed that $\Omega$, the count of the number of prime factors with multiplicity), should have infinitely many barriers. Selfridge found the largest barrier for $\Omega$ which is $<10^5$ is $99840$.
In [ErGr80] this problem is suggested as a way of showing that the iterated behaviour of $n\mapsto n+\omega(n)$ eventually settles into a single sequence, regardless of the starting value of $n$ (see also [412] and [414]).
Erdős and Graham report it could be attacked by sieve methods, but 'at present these methods are not strong enough'.
A more precise result was proved by Hall and Tenenbaum [HaTe88] (see Section 4.6), who showed that the upper density is $\ll\epsilon \log(2/\epsilon)$. Hall and Tenenbaum further prove that $\tau^+(n)/\tau(n)$ has a distribution function.
Erdős and Graham also asked whether there is a good inequality known for $\sum_{n\leq x}\tau^+(n)$. This was provided by Ford [Fo08] who proved \[\sum_{n\leq x}\tau^+(n)\asymp x\frac{(\log x)^{1-\alpha}}{(\log\log x)^{3/2}}\] where \[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\]
In [Er79] Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggets that \[\lim_{n\to \infty}\max_{m<n}(\tau(m)+m-n)=\infty.\]
In [Er79d] Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which \[\max_{n-k<m<n}(m+\tau(m))\leq n+2,\] but 'this is hopeless with our present methods', although it follows from Schinzel's Hypothesis H.
See also [413].
More generally, Brun's sieve can be used to prove that if $B\subseteq \mathbb{N}$ is a set of pairwise coprime integers with $\sum_{b<x}\frac{1}{b}=o(\log\log x)$ then $A=\{ n: b\nmid n\textrm{ for all }b\in A\}$ has the translation property. Erdős did not know what happens if the condition on $\sum_{b<x}\frac{1}{b}$ is weakened or dropped altogether.
What if the condition that $p$ is prime is omitted? Selfridge and Wagstaff made a 'preliminary computer search' and suggested that there are infinitely many $n$ not of this form even without the condition that $p$ is prime. It should be true that the number of exceptions in $[1,x]$ is $<x^c$ for some constant $c<1$.
Most generally, given some infinite set $A\subseteq \mathbb{N}$ and function $f:A\to \mathbb{N}$ one can ask for sufficient conditions on $A$ and $f$ that guarantee every large number (or almost all numbers) can be written as \[am^2+b\] for some $m\in A$ and $a\geq 1$ and $0\leq b<f(m)$.
In another direction, one can ask what is the minimal $c_n$ such that $n$ can be written as $n=ap^2+b$ with $0\leq b<c_np$ for some $p\leq \sqrt{n}$. This problem asks whether $c_n\leq 1$ eventually, but in [Er79d] Erdős suggests that in fact $\limsup c_n=\infty$. Is it true that $c_n<n^{o(1)}$?
Is it true that for all $m\geq n+k$ \[M(n,k) \neq M(m,k)?\]
In general, how many solutions does $M(n,k)=M(m,l)$ have when $m\geq n+k$ and $l>1$? Erdős expects very few (and none when $l\geq k$).
The only solutions Erdős knew were $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$.
In [Er79d] Erdős conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\geq n+k$ then $\prod_{i\leq k}(n+i)$ and $\prod_{i\leq k}(m+i)$ cannot have the same set of prime factors.
Let $k\geq 3$. Are there infinitely many $m,n$ with $m\geq n+k$ such that \[M(n,k)>M(m,k+1)?\]
The answer is yes, as proved in a strong form by Cambie [Ca24].
It is easy to see that there are infinitely many solutions to $M(n,k)>M(m,k)$. If $n_k$ is the smallest $n$ with this property (for some $m$) then are there good bounds for $n_k$? Erdős writes that he could prove $n_k/k\to \infty$, but knew of no good upper bounds.
Erdős also asked the following: If $u_k$ is minimal such that $M(u_k,k)>M(u_k+1,k)$ and $t<\min(u_k,T)$ then is it true that $M(t,k)\leq M(T,k)$? Stijn Cambie and Wouter van Doorn have observed that there are many counterexamples to this with $t=u_k-1$ and $T=u_k+1$. For example, when $k=7$ we have $u_k=7$, yet $M(6,7)=M(7,7)>M(8,7)$.
See also [677].