17 Consecutive Integers each with Common Factor with Product of other 16

Theorem
The $17$ consecutive integers from $2184$ to $2200$ have the property that each one shares a common divisor greater than $1$ with the product of the other $16$.

Proof
We obtain the prime decomposition of all $17$ of these integers:

It can be seen by inspection that each of the integers in this sequence shares at least one prime factor with at least one other.

It is then worth noting that:

and it can be seen that the sequence can be extended neither upwards nor downwards.