The abundance of a positive integer 
, sometimes also called the abundancy
(a term which in this work is reserved for a different but related quantity), is
the quantity
 |
where 
is the divisor
function. The abundances of 
, 2, ... are 
, 
,
, 
, 
,
0, 
, 
, 
,
, 
, 4, 
, 
,
, 
, ... (OEIS A033880).
The following table lists special classifications given to a positive integer 
based on the value of 
.
In the almost perfect row, the listed powers of 2 are known
examples. It remains unknown whether every almost
perfect number is a power of 2.
 | class | OEIS | list of  |
 | deficient number | A005100 | 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, ... |
 | almost perfect number | A000079 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ... |
| 0 | perfect number | A000396 | 6, 28, 496, 8128, ... |
| 1 | quasiperfect number | none known | |
 | abundant number | A005101 | 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, ... |
Values of 
such that 
is odd are given by 
, 2, 4, 8, 9, 16, 18, 25, 32, ... (OEIS A028982;
i.e., the union of nonzero squares
and twice the squares). Values of 
such that 
is a square number are
given by 
,
12, 28, 70, 88, 108, 168, ... (OEIS A109510).
