$$ \begin{align} 2254 & = 2\cdot7\cdot7\cdot23 \\ 2255 & = 5\cdot11\cdot41 \\ 2256 & = 2\cdot2\cdot2\cdot2\cdot3\cdot47 \\ 2257 & = 37\cdot61 \\ 2258 & = 2\cdot1129 \\ 2259 & = 3\cdot3\cdot251 \\ 2260 & = 2\cdot2\cdot5\cdot113 \\ 2261 & = 7\cdot17\cdot19 \\ 2262 & = 2\cdot3\cdot13\cdot29 \\ 2263 & = 31\cdot73 \\ 2264 & = 2\cdot2\cdot2\cdot283 \\ 2265 & = 3\cdot5\cdot151 \\ 2266 & = 2\cdot11\cdot103 \end{align} $$ I this question I pointed something out, of which the foregoing is another instance.
The integer parts of the square roots of all of these numbers are equal to $47$, which is the $15$th prime, so if we're checking for primality, we need to search that far and the rest does itself. Now notice which primes $\le47$ appear above: $$ 2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29,\,31,\,37,\,41,\,\bullet,\,47 $$ i.e. all of those first $15$ except $43$. Hence nearby numbers can be divisible only by small primes that recur frequently or primes bigger than the square root of the number being factored. (In particular $2272$ is $71$ times a power of $2$, and $2268$ has no prime factors bigger than $7$.)
Is there any reasonable sense it which it could be said that we shouldn't find it so surprising that this---all those early primes occurring so close together---occurs among numbers as small as these? Should we expect frequent instances of this?
PS: "Small" should probably be taken to mean these numbers are not much bigger than $47^2=2209$, where, remember, $47$ is the biggest prime number $\le$ the square roots of these numbers.