A binary coin

Discrete Mathematics Level pending

You have a fair coin with a "1" on one side and a "0" on another side. You decide to flip it until you get a string of "1"s and "0"s that is the binary representation of a number you are thinking of.

e.g. If you pick the number 3, then its binary representation would be "11" and the expected number of flips to get this combination would be 6.

Now for the question:

What is the smallest integer, \(n\), such that the expected number of flips to get its binary representation is greater than 40

Note: Leading zeros aren't allowed. i.e. The number 5 would be represented by 101, not 0101 nor 00101.

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