Image of the Euler phi function

Question

I was reading a text about arithmetic functions, which ofcourse mentioned the Euler phi function. I was wondering whether $\phi(n)$ takes on all positive integer values. The answer doesn't seem so simple because the Euler phi function behaves in a very odd way.

Answer 1

No, $\phi(n)$ does not attain all (even) positive integer values, and the the list of even numbers which are not attained can be found here: http://oeis.org/A005277. It starts with $$ 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, $$ Of course, odd values $>1$ are never attained, because $\phi(n)$ has a factor $2$ if $n\ge 3$, because $\phi$ is multiplicative and $\phi(p^n)=p^n-p^{n-1}$ is even for an odd prime $p$, as well as $\phi(2^n)=2^n-2^{n-1}$ for $n\ge 2$.