What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2$?

Let

$$I(x) = \frac{\sigma(x)}{x}$$

be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example,

$$\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.$$

My question is this: What proportion of the positive integers satisfy the inequality $$I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2?$$

Note that we necessarily have $n > 1$ from the left-hand inequality.

(A similar question is posted in MSE here.)

-
What is the "abundancy index"? In particular, what is $\sigma(x)$? And why did you post on MO within 7 minutes of posting on MSE instead of waiting for responses there? –  Vidit Nanda Sep 9 '13 at 15:39
It seems likely that the asymptotic density is 0 using standard arguments. What have you tried? –  The Masked Avenger Sep 9 '13 at 16:39
@ViditNanda, my short answer to your last question is that I am not so sure if this question can be considered a "research-level" question as per MO standards. –  Jose Arnaldo Dris Sep 9 '13 at 16:57
@TheMaskedAvenger, I've tried polynomial division on the middle inequality, thereby getting the final result $$2n^2 - 2n + 1 \leq \sigma(n^2) < 2n^2.$$ This means that the abundance $a(n^2) = \sigma(n^2) - 2n^2$ satisfies $1 - 2n \leq a(n^2) < 0$, while the deficiency $d(n^2) = 2n^2 - \sigma(n^2)$ satisfies $0 < d(n^2) \leq 2n - 1$. –  Jose Arnaldo Dris Sep 9 '13 at 17:03
@StevenLandsburg : Thank you for pointing that out, I certainly did not mean to be rude. I am just another amateur MO/MSE OP looking for an answer to my question. –  Jose Arnaldo Dris Sep 10 '13 at 4:18

The function $I(n^2)$ has a continuous limiting distribution (look up the Erd\H{o}s--Wintner theorem). Since your lower limit $2n/(n+1)$ converges to your upper limit of $2$ as $n\to\infty$, the continuity of the distribution function shows that the limiting proportion of $n$ satisfying your inequality is $0$.
I forgot to ask (and my apologies in advance if this is too dumb a question), but does your result imply that the asymptotic density or limiting proportion of $n$ satisfying the inequality $$I(n^2) < \frac{2n}{n + 1},$$ is $1$? –  Jose Arnaldo Dris Sep 17 '13 at 10:03