I wondered if one can to get easily an answer for the following question (I have thought about the other direction $\Leftarrow$). I don't know if it is in the literature, please refer it in comments in such case.
We denote the Euler's totient function as $\varphi(m)$, the Wikipedia encyclopedia has the article Euler's totient function.
Conjecture. Let $n\geq 1$ be an integer satisfying the identity $$\varphi(n)=\frac{1}{2n}\sum_{1\leq d\mid n}\varphi(dn),\tag{1}$$ then $n$ is an even perfect number.
Due that it isn't unknown if there exists odd perfect numbers I don't know if my question (the following question) is difficult, using elementary computations from $(1)$. It is known that the Euler's totient function is multiplicative and I know also properties of multiplicative functions and Möbius inversion.
Question. Is it true previous conjecture? Many thanks.
My only idea is that maybe can be useful to use the representation of the Euler's totient function as a Dirichlet convolution. If my conjecture is known from the literature feel free to answer as a reference request, and I try to search an read the statement from the literature.
With a Pari/GP script (or similar than these that I can to evaluate)
for(n=1,100000,if(2*n*eulerphi(n)==sumdiv(n,d,eulerphi(d*n)),print(n)))
you can to see that there aren't small counterexamples, just copy and paste previous code in the web Sage Cell Server and choose as language GP to evaluate it.