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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 $2\cdot 10^8$ .

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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

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