Let us call $k$-tuple $(n+1,n+2,..,n+k)$ a *$k$-factorazy tuple* if we have $p(n+1)<p(n+2)<...<p(n+k)$ where $p(m)$ denotes biggest prime factor of $m$.

I would like to know:

> Is it true that for every $k \in \mathbb N \setminus \{1\}$ there exists at least one *$k$-factorazy tuple*? If not, what is the maximal value of $k$? For which $k$´s there exist an infinite number of *$k$-factorazy tuples*? What is known about this topic?