'imperfect' numbers A perfect number (integer) is equal to the sum of its divisors, including 1, and excluding itself. This has been around since Euclid.
Recently, I noticed that at least for the initial integers, it is more common for that sum of divisors to be smaller than the number in question.
However, for example, using the same rules the sum of the divisors of 12 is actually sixteen: The sum is here greater than the whole. 
Clearly, not perfect. Therefore, imperfect. But, has anyone studied these numbers?

The sum of all proper divisors of a number is less than, equal to, or greater than the number,
according as the number is deficient, perfect, or abundant. 
The first 28 abundant numbers are $12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60,$ 
$66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104,$ $ 108, 112, 114, $ and $120.$ 
They are sequence A005101 in the On-Line Encyclopedia of Integer Sequences.  
The smallest odd abundant number is $945$.  
Every multiple (beyond $1$) of a perfect number is abundant.  
You can read more about these numbers on Wikipedia.
