Abundance

The abundance of a number $n$ is the quantity

$$A(n)\equiv\sigma(n)-2n,$$

where $\sigma(n)$ is the Divisor Function. Kravitz has conjectured that no numbers exist whose abundance is an Odd Square (Guy 1994).

The following table lists special classifications given to a number $n$ based on the value of $A(n)$.

$A(n)$	Number
$<0$	Deficient Number
$-1$	Almost Perfect Number
0	Perfect Number
1	Quasiperfect Number
$>0$	Abundant Number

See also Deficiency

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-46, 1994.