Suppose , $k$ is a positive integer and there exists a positive integer $n$ with $\varphi(n)=k$
- Can the smallest value $n$ with $\varphi(n)=k$ be even ?
- Can all values $n$ with $\varphi(n)=k$ be even ?
I did not find a counterexample even for the first statement upto $3\cdot 10^4$. It is clear that the smallest value $n$ cannot be of the form $4m+2$ because then $\frac{n}{2}$ would also be a possible value.
The given statement is a strengthening of the conjecture that there is no $k$ such there is exactly one $n$ with $\varphi(n)=k$.
