Too Many 2s

Every positive integer can be written as a sum of distinct powers of two. For example, \(101={ 2 }^{ 0 }+{ 2 }^{ 2 }+{ 2 }^{ 5 }+{ 2 }^{ 6 }\)

As we can see, \(101\) is written as a sum of four distinct powers of two. How many positive integers less than \({ 2 }^{222}\) can be written as a sum of four distinct powers of two?

Note: \({ 38=2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 4 }+{ 2 }^{ 4 }\) is NOT valid, since \({ 2 }^{ 4 }\) appears twice.

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