In this post, for integers $l\geq 1$, we denote the Euler's totient function as $\varphi(l)$ and the sum of divisors function as $\sigma(l)=\sum_{d\mid l}d$. An even number $n$ is said to be an even perfect number if satisfies $$\sigma(n)=2n.$$This is the MathWorld's article dedicated to Perfect Number. In this post we show an equation that satisfies every even perfect number greater than $6$, and after we ask about a related conjecture.
Claim. Let $n>6$ an even perfect number (thus $\gcd(3,n)=1$). Then $$\sigma\left(3n+2\varphi(n)-\varphi(n+\sigma(n))\right)=8\cdot(n-\varphi(n+\sigma(n)))\cdot(\sigma(n)-2\varphi(n+\sigma(n))-1).$$ Sketch of proof. Each even perfect number $n>6$ satisfies the equations $$\varphi(n+\sigma(n))=n-\frac{1+\sqrt{1+8n}}{4},\tag{1}$$ and $$\sigma\left(2n+2\varphi(n)+\frac{1+\sqrt{1+8n}}{4}\right)=8n.\tag{2}$$ And from here it's possible to prove that satisfies the required equation. $\square$
I would like to know what work can be done about the other implication, that is what about next conjecture.
Conjecture. Let $m\geq 1$ a positive integer satisfying $$\sigma\left(3m+2\varphi(m)-\varphi(m+\sigma(m))\right)=8\cdot(m-\varphi(m+\sigma(m)))\cdot(\sigma(m)-2\varphi(m+\sigma(m))-1),$$ then $m$ is an even perfect number.
Question. Although the equation that I created is artificious, I would like to know if can be done some work about it (I'm curious about that because I think we can deduce very little or maybe nothing). What can be deduced (what calculations/reasonings) about the veracity of previous Conjecture? Many thanks.
Computational fact. If my program is right the Conjecture holds for $1\leq m\leq 10^5$.