Consider the set of integers between $2$ and some large number $N$. For each such number $n$, we can compute the quantity $x = GPF(n)/\sqrt{n}$, where $GPF(n)$ is the greatest prime factor function. $x$ is a real number which can be greater than, less than, or equal to $1$. It is bounded by $2/\sqrt{n} \leq x \leq \sqrt{n}$, where the lower bound occurs when $x$ is a power of two, and the upper bound occurs when $x$ is prime. $x=1$ only when $n$ is the square of a prime.
Now suppose we take all the values of $x$ for all the numbers $n$ between $2$ and $N$. This data can then be binned and turned into an estimate of the density function $f(x)$ for $x$. I've computed this density function for $N = {10}^7$ using Mathematica, and it is plotted below on a log-log scale.
There's a discontinuity at $x=1$, presumably due to numbers which are squares of primes. The really interesting part, however, is in the region $x>1$, where $f(x)$ appears to fall on a straight line in the log-log plot. This implies that $f(x)$ obeys a power law for large $x$. Fitting the data for $x>1$ to a power law form yields a very good fit of $f(x) \propto 1/x^p$, where $p\approx 1.113$. For sufficiently large $N$, this exponent $p$ appears to be independent of $N$. I tried it for $N = {10}^5$, $N = {10}^6$, and $N = {10}^7$ and got about the same value for $p$ each time.
My questions are: What is this number $p\approx 1.113$? Where does it come from? Is it related to known constants? Why does $f(x)$ obey a power law at all?
Edited to add: The last part of the Mathworld entry on the greatest prime factor function seems relevant.
Edit 2: This previous math.stack discussion also seems very relevant.