Proper Divisor
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.
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