Let us call $k$-tuple $(n+1,n+2,..,n+k)$ a $k$-factorazy tuple if we have $p(n+1)<p(n+2)<\cdots<p(n+k)$ 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?

Let us call $k$-tuple $(n+1,n+2,..,n+k)$ a factorazy $k$-tuple if we have $p(n+1)<p(n+2)<\cdots<p(n+k)$ 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 factorazy $k$-tuple? If not, what is the maximal value of $k$? For which $k$´s there exist an infinite number of factorazy $k$-tuples? What is known about this topic?

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?

Let us call $k$-tuple $(n+1,n+2,..,n+k)$ a $k$-factorazy tuple if we have $p(n+1)<p(n+2)<\cdots<p(n+k)$ 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?