# 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