Consider the divisors of $n$, $$d_1 = 1, d_2, d_3, ..., d_r=n$$ in ascending order and $r \equiv r(n)$ is the number of divisors of $n$.
Is there any expression $f(n) < r(n)$ such that, $$\sum_{k=1}^{r-1} \frac{d_k}{d_{k+1}} < f(n)$$
Or equivalently, in a generalized way, is there any expression $\phi_a(n) < \sigma_a(n)$ such that, $$\sum_{k=1}^{r-1} \frac{d_k^{a+1}}{d_{k+1}} < \phi_a(n)$$ for positive integers $a$ ?
PS. Looking for a study on tight bounds over these sums of ratios. Any references will be greatly appreciated.