Sum of digits equation
Public  07/08/16  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 $nx_i$
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.
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 $nx_i$
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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