I posted a version of this to stackexchange and got 12 up-votes and no answers in somewhat more than a day. Someone in a comment construed it as asking for a lot of novel research including figuring out how to make the statement precise. But the actual question was whether someone had already done those things.

I was looking at this set of prime factorizations:
$$
\begin{align}
1100 & = 2\times2\times5\times5\times11 \\
1101 & =3\times 367 \\
1102 & =2\times19\times29 \\
1103 & =1103 \\
1104 & = 2\times2\times2\times2\times 3\times23 \\
1105 & = 5\times13\times17 \\
1106 & = 2\times7\times79
\end{align}
$$
I noticed that **all** of the first 10 prime numbers, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, appear within the factorizations of only seven consecutive integers (and within only six of them). (The next prime number, 31, has its multiples as far from 1100 as it could hope to get them (1085 and 1116).) So no nearby number could hope to be divisible by 29 or 23, nor even by 7 for suitably adjusted values of "nearby". Consequently when you're factoring nearby numbers, you're deprived of those small primes as potential factors by which they might be divisible. So nearby numbers, for lack of small primes that could divide them, must be divisible by large primes. And accordingly, not far away, we find $1099=7\times157$ (157 doesn't show up so often---only once every 157 steps---that you'd usually expect to find it so close by) and likewise 1098 is divisible by 61, 1008 by 277, 1096 by 137, 1112 by 139, 1095 by 73, 1094 by 547, etc.; and 1097 and 1109 are themselves prime.

So if an unusually large number of small primes occur unusually close together as factors, then an unusually large number of large primes must also be in the neighborhood.

Are there known precise results quantifying this phenomenon?