# Not A Perfect Number

**Number Theory**Level 3

\[\begin{array}{ll} 6 = 1 + 2 + 3 & 28 = 1 + 2 + 4 + 7 + 14 \\ 6 = 2^1\cdot (2^2 - 1) & 28 = 2^2\cdot (2^3 - 1) \end{array}\]

6 and 28 are perfect numbers, because each of them is equal to the sum of its proper divisors, as shown above. They are also numbers of the form \(2^n\cdot (2^{n+1} - 1)\).

Not all numbers of the form \(2^n\cdot (2^{n+1}-1)\) are perfect numbers. Let's call those numbers *imperfect*. For instance, 120 is an imperfect number because
\[120 = 2^3\cdot (2^4-1)\]
yet
\[120 \not= 1+2+3+4+5+6+8+10+12 \\ +15+20+24+30+40+60.\]

What is the smallest imperfect number greater than 120?