For a bounded range of positive integers $n,n+1,\ldots,m,$ call a prime number "small" if it does not exceed $\sqrt m,$ so that if one is trying to factor all of these numbers into primes, one need only check "small" primes, and it those don't finish off the factorization, one does one last division, getting a "large" prime.
Now suppose the range is the gap between the $k\text{th}$ and $(k+1)\text{th}$ primes. For large values of $k,$ it is unusual to find all of the small primes as factors of numbers within the gap. However, it does happen with the gap between $1327$ and $1361.$ The small primes in that case are the first $11$ primes: $2,3,5,7,11,13,17,19,23,29,31.$ All of those are factors of some number in that gap.
I suspect that somewhere in the literature it is proved that for sufficiently large $k,$ this never happens. That raises the question of how large is sufficiently large. I won't be surprised if this particular gap is the very last one for which it happens. (Nor will I be greatly surprised if it is not.)
If there are indeed only finitely many such gaps, is there a list of them somewhere? Maybe on OEIS?
Best Answer
As noted by Will Jagy in the comments, this is closely related to the size of prime gaps: Any gap of size at least $\sqrt{m}$ has this property.
In fact, every gap with this property has size at least $\sqrt{m}/\log m$: We have $$\prod_{i=n}^{m } i\geq \prod_{\substack{p \leq \sqrt{m}\\ \textrm{prime}}} p$$ and the RHS has size $e^{\sqrt{m}}$ by the prime number theorem while the left hand size $\leq m^{m+1-n}$ so we must have $(m+1-n) \gtrapprox \frac{\sqrt{m}}{\log m}$.
So the conjecture that finitely many of these exist is stronger than the statement that there are finitely many prime gaps of size $\sqrt{m}$ but weaker than the statement that there are finitely many prime gaps of size $\leq (1-o(1))\frac{\sqrt{m}}{\log m}$. It's certainly conjecturally true as even stronger bounds are conjectured, but beyond the reach of proof.
From my point of view the remaining thing to do is to shrink the gap between the minimum gap size required for every small prime to appear and the minimum gap size which guarantees that all small primes appear.
To that end, observe that for $p\leq \sqrt{m}$ for the gap to have size $\leq m-n$, we must have $ \left\{ \frac{m}{p}\right\} \leq \frac{m-n}{p}$ so if $p_1 \leq p_2$, the distance from $\frac{m}{p_1} - \frac{m}{p_2} = \frac{m (p_1-p_2)}{p_1p_2} $ to the nearest integer is at most $\frac{m-n}{p_1}$.
The results of Zhang and Maynard on bounded gaps between primes count pairs of primes with bounded gaps in dyadic intervals. I think the same methods without modification count pairs of primes with bounded gaps in slightly smaller intervals, so we can choose $p_1= (1+o(1)) \sqrt{ \frac{m}{\sqrt{2}}} $ so that $p_2-p_1=O(1)$. Then $\frac{ m(p_1-p_2)}{p_1p_2}$ will be $1+o(1)$ times a bounded integer times $\sqrt{2}$, so its distance from the nearest integer will be bounded away from $0$ by an absolute constant, so $\frac{m-n}{p_1}$ must be greater than an absolute constant, and thus the gap must be greater than an absolute constant times $\sqrt{m}$.