A positive integer $n$ is called $k$-rough if all of the prime factors of $n$ strictly exceed $k$.
For $k$ fixed and $n$ large, what's the shortest interval $[n,n+t]$ that contains a $k$-rough number ?



Rough numbers in short interval
Being $k$-rough is the same as being relatively prime to the product of primes up to $k$. Hence if $P(k)$ denotes that product, and $j$ denotes the [Jacobsthal function](https://oeis.org/wiki/Jacobsthal_function), then $t=j(P(k))-1$ works for every $n$, while $t=j(P(k))-2$ fails for infinitely many $n$'s. In particular, it follows from a result of [Iwaniec (1971)](http://matwbn.icm.edu.pl/ksiazki/aa/aa19/aa1911.pdf) that $t=ck^2$ works for some absolute constant $c>0$.


