Two kinds of prime gaps $$1361 - 1327 = 34$$
Between these two prime numbers there are no others. No prime gaps this big come before this one; i.e. this one is "maximal".
The largest prime not exceeding the square roots of any of them is $31.$ All prime numbers not exceeding $31$ divide some number between these two primes.
$$ 9587 - 9551 = 36 $$
Between these two prime numbers there are no others. No prime gaps this big come before this one, i.e. this one is "maximal".
The largest prime not exceeding the square roots of any of them is $97.$ Some prime numbers not exceeding $97$ divide no number between these two primes. In fact, $97$ is one of those. So are $83$ and $89.$
Are infinitely many maximal prime gaps like the first one mentioned above, in this respect? How are they distributed among maximal prime gaps? (I'd guess they occur less frequently than those like the second one.)
 A: Consider a maximal gap that follows a very large prime $p$. 


*

*Maximal gap sizes are $O(\log^2 p)$ (by  Cramer's conjecture). 

*Almost all maximal gaps are asymptotically equivalent to $\log^2 p$ (a modified form of Shanks conjecture).

*Hence, there are $O(\log^2 p)$ composite integers inside our maximal gap.

*The expected number of distinct prime factors of one such composite integer is $\sim\log\log p$ (by the Erdos-Kac theorem).

*Therefore the expected total number of distinct prime factors dividing at least one integer inside the gap is $f(p)=O(\log\log p \cdot \log^2 p)$

*On the other hand, $\pi(\sqrt{p})\sim {\sqrt{p}\over\log \sqrt{p}}$ (by the prime number theorem).

*Thus $f(p) = o(\pi(\sqrt{p}))$, i.e. we expect only a zero proportion of primes below $\sqrt{p}$ to be factors of some integer inside the gap.
On probabilistic grounds we conclude that gaps of the second type are much more common. In fact we should expect that gaps of the first type are only a zero proportion of all maximal gaps. 
Moreover, we should expect only finitely many gaps of the first type.
This is provable if Cramer's conjecture is true. Let $\Pi$ be the product of all  composites inside the gap. What's the order of magnitude of $\log\Pi$? Firstly, $\Pi$ is a product of $\sim\log^2⁡p$ integers each of size $\sim p$; so
$$
\log\Pi \sim \log(p^{\log^2⁡p}) \sim \log^2⁡p \cdot \log p = \log^3⁡p.
$$
 On the other hand, $\Pi$ is a product of all prime factors of these composites. By assuming that all primes below $\sqrt{p}$ participate in the product (and small primes participate many times) we would get a different asymptotic size for the same product: 
$$
\log\Pi > \log(\sqrt{p}\#) \sim \sqrt{p} \quad (\# \mbox{ denotes primorial).}
$$
But $$\lim_\limits{p\to\infty}{\log^3 p \over \sqrt{p}}=0.$$
That's a contradiction -- unless there are only finitely many examples of a maximal gap near $p$, containing composites divisible by every prime below $\sqrt{p}$.
A: The short story is the divisor is larger than the gap and this may true for other maximal gaps.  The rest of the story:  $9587 / 83 = 115.5$ and the next prime number is $113$ or $127.$  $83 \times 127 = 10541$ or $83 \times 113 = 9379.$  $9587 / 89 = 107.7$ the next prime numbers are $107$ or $109.$  $9587 / 97 = 98.8$ and the next prime numbers are $97$ or $101.$  These calculations show how they were not related to the gap of $34$ between $9551$ and $9587.$
