Consider an integer \(n\) in base 10, whose binary form can be expressed as a string \(S\). Let the minimum value of \(n\) be \(N\), such that the sub-strings of \(S\) include binary numbers from \(1\) to \(10000_2\). What is the sum of digits of \(N\)?

**Details and Assumptions**

- For example, the integer \(9\), whose binary form can be expressed as a string \(1001\) includes sub-strings of \(0,1,10,1001\), but not \(101\).

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