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Problem #340

Sum of digits equation
 Public ★(x8) 07/08/16 by Philippe_57721 10xp Programming 88.9%

We define $DS(n) = n + SOD(n)$ where SOD(n) is the sum of the digits of n (See problem 260)

It can be proved that $10^{13}+1$ is the smallest integer such as the equation $DS(x) = n$ has 3 solutions.
$$\begin{equation*} 10^{13}+1= \begin{cases} DS(9999999999892) \\ DS(9999999999901) \\ DS(10000000000000) \end{cases} \end{equation*}$$ It can be proved too that $n = 10^{2222222222224}+10000000000002$ is the smallest integer with 6 solutions.

Obviously, if x is a solution we must have $x \lt n$. Thus, we can write a solution as $n-x_i$

Find these solutions.

Answer format: $x_1,x_2,x_3,x_4,x_5,x_6$ // $x_i$ in ascending order

Example : 1,100,109 // For $10^{13}+1$

[My timing: < 1 sec]

Thanks to sinan who helped me to improve this problem.

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