All even numbers $n$ with $\sigma_1(n)=2n-8$ have digital root $4$

How to prove or disprove following claim :

Let $n$ be an even natural number such that $\sigma_1(n)=2n-8$ . All numbers $n$ with this property have digital root $4$ .

I have tested this statement for all $n$ below $2\cdot 10^8$ .

Solutions Collecting From Web of "All even numbers $n$ with $\sigma_1(n)=2n-8$ have digital root $4$"

This is not answer, just a little context:

The quantitiy $2n-\sigma_1(n)$ is called the “deficiency” of $n$. The first numbers with deficiency $8$ are
$$
\begin{align}
22 &= (2)(11) \\
130 &= (2)(5)(13) \\
184 &= (2)^3(23) \\
1012 &= (2)^2(11)(23) \\
2272 &= (2)^5(71) \\
18904 &= (2)^3(17)(139) \\
33664 &= (2)^7(263) \\
70564 &= (2)^2(13)(23)(59) \\
85936 &= (2)^4(41)(131) \\
100804 &= (2)^2(11)(29)(79) \\
391612 &= (2)^2(13)(17)(443) \\
527872 &= (2)^9(1031) \\
1090912 &= (2)^5(73)(467) \\
17619844 &= (2)^2(11)(37)(79)(137) \\
\end{align}
$$

a couple more terms (up to $1661355408388$, all of which have digital root $4$) can be found at http://oeis.org/A125247