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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 $\varphi(n)=k$.

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See OEIS sequence A002181. The answers to both questions are yes. For example $\varphi(n)=16842752$ only for even $n$.

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