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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.

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    I added the reference-request tag to your question due to your post-script.2011-12-30

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