How can I construct a number $n$, such that $gcd(n+k,100!)\ne 1$ for all $k=0,...,256$

Question

Here :

https://oeis.org/search?q=2%2C4%2C6%2C10%2C14%2C22%2C26%2C34%2C40%2C46&sort=&language=german&go=Suche

it is indirectly claimed that there exists a number $n$, such that $n+k$ has a prime factor $p\le 97$ for $k=0,1,2,...,256$.

Despite great effort, I could not find such a number.

How can I construct such a number ?

What is the smallest number $n$ doing the job ?

How can I show that at least one of the numbers $n,n+1,...,n+257$ is coprime to $100!$ for every $n$ ?