A positive proper divisor is a positive divisor of a number 
,
excluding 
itself. For example, 1, 2, and 3 are positive proper divisors of 6, but 6 itself
is not. The number of proper divisors of 
is therefore given by
 |
where 
is the divisor function. For 
, 2, ..., 
is therefore given by 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, ...
(OEIS A032741). The largest proper divisors
of 
, 3, ... are 1, 1, 2, 1, 3, 1, 4, 3,
5, 1, ... (OEIS A032742).
The term "proper divisor" is sometimes used to include negative integer divisors of a number 
(excluding 
).
Using this definition, 
,
, 
, 1, 2, and 3 are the proper divisors of 6, while 
and 6 are the improper divisors.
To make matters even more confusing, the proper divisor is often defined so that 
and 1 are also excluded. Using this
alternative definition, the proper divisors of 6 would then be 
, 
, 2, and 3, and the improper
divisors would be 
,
, 1, and 6.
