Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$.


Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. 


Is $\limsup\_{n\rightarrow\infty} \frac{f(n)}{n} > 0$. It is less than $\frac{1}{2}$ by my previous statement, but I don't know how to proceed.



Probability that a positive integer is the euler phi function of another positive integer
See Erick Wong's response [here](https://math.stackexchange.com/questions/291334/values-taken-by-eulers-phi-function). In particular, Kevin Ford proved (in more precise form) that
$$ f(n) = \frac{n}{\log n} \exp\left(O(\log \log \log n)^2\right),$$
whence $f(n)/n$ tends to zero. The same consequence also follows from an earlier result of Pillai (1929), available online [here](http://www.ams.org/journals/bull/1929-35-06/S0002-9904-1929-04799-2/S0002-9904-1929-04799-2.pdf).


