Too Many 2s

Every positive integer can be written as a sum of distinct powers of two. For example, 101=20+22+25+26101={ 2 }^{ 0 }+{ 2 }^{ 2 }+{ 2 }^{ 5 }+{ 2 }^{ 6 }

As we can see, 101101 is written as a sum of four distinct powers of two. How many positive integers less than 2222{ 2 }^{222} can be written as a sum of four distinct powers of two?

Note: 38=21+22+24+24{ 38=2 }^{ 1 }+{ 2 }^{ 2 }+{ 2 }^{ 4 }+{ 2 }^{ 4 } is NOT valid, since 24{ 2 }^{ 4 } appears twice.

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