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.
[Edit: September 10 2013] This question has already been answered by Don in MO here.  Thanks everyone! [End edit]

This is extracted from my answer to
Does this inequality hold true, in general?
Apply the result stated below
to get the results
for any constraints whatsoever.
I did a Google search for
"density of euler phi function".
The second link is
http://www.ams.org/journals/proc/2007-135-09/S0002-9939-07-08771-0/S0002-9939-07-08771-0.pdf.
This paper,
by ANDREAS WEINGARTNER,
is titled
"THE DISTRIBUTION FUNCTIONS OF σ(n)/n AND n/ϕ(n)".
Here is its abstract:
"Let σ(n) be the sum of the positive divisors of n. We show that
the natural density of the set of integers n satisfying σ(n)/n ≥ t is given
by 
$\exp\big(−e^{t e^{−γ}(1 + O(t^{−2}))}\big)$
, where γ denotes Euler’s constant. The same
result holds when σ(n)/n is replaced by n/ϕ(n), where ϕ is Euler’s totient
function."
