Let \(f\) be a function in the natural numbers, such that \(f(n)\) gives the binary representation of \(n\) in base 10. For example, \(f(5) = 101\), because 5 is represented as 101 in binary.

A number \(n\) is called *good* if \(n\) and \(f(n)\) are both prime numbers. For example, 5 is good, because 5 and 101 are prime numbers. 7 is not good, since \(f(7) = 111 = 3 \times 37\) is not prime.

The smallest good number is 3. The second smallest good number is 5. Find the \(19^\text{th} \) smallest good number.

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