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.

REFERENCES:

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 8-9, 2004.

Sloane, N. J. A. Sequences A032741 and A032742 in "The On-Line Encyclopedia of Integer Sequences."