If $1=d_1<\cdots <d_{\tau (n)}=n$ are the divisors of $n$, then let ${\tau }_{\perp }(n)$ count the number of $i$ for which $(d_i,d_{i+1})=1$.Is it true that ${\tau }_{\perp }(n)/\omega (n)\to \infty$ for almost all $n$? Is it true that$${\tau }_{\perp }(n)<\exp ((\log n{)}^{o(1)})$$for all $n$?Let$$g(k)=\underset{\omega (n)=k}{\max }{\tau }_{\perp }(n),$$where $\omega (n)$ counts the number of distinct prime divisors of $n$, and $n$ is restricted to squarefree integers. Determine the growth of $g(k)$.