To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?
By the uniqueness of the binary representation of positive integers, there is only one way to represent 1562 as a sum of distinct powers of $2$.  To find this representation, we convert 1562 to binary form.  The largest power of $2$ less than 1562 is $2^{10}=1024$.  The difference between 1024 and 1562 is $538$. The largest power of 2 less than 538 is $2^9=512$.  The difference between 538 and 512 is 26.  The largest power of 2 less than 26 is $2^4=16$, which leaves us with $26-16=10$. Continuing the process, we get $2^3=8$ and $2^1=2$. So, we have found that $1562=2^{10}+2^9+2^4+2^3+2^1$.  The sum of the exponents of 2 in this representation is $\boxed{27}$.