An integer is said to be an even perfect number if satisifies $\sigma(n)=2n$, where $\sigma(n)$ is the sum of the positive divisors of $n$. The first few even perfect numbers are $6,28,496$ and $8128$.
Question. I wondered if we can prove or refute the following statement:
Let $\varphi(n)$ the Euler's totient function and we denote the product of the distinct primes dividing a natural $n>1$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ (it is this arithmetic function from the Wikipedia Radical of an integer). If $n$ satisfies the equation $$\varphi(n)=\left(\frac{1+\sqrt{1+8n}}{8}\right)\cdot\left(\operatorname{rad}(n)-\frac{1+\sqrt{1+8n}}{2}\right)$$ then $n$ is an even perfect number.
Many thanks.