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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.

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up vote 14 down vote accepted

See Erick Wong's response here. 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.

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