Abundance

The abundance of a number , 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 number  based on the value of .

	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 square are given by , 12, 28, 70, 88, 108, 168, ... (OEIS A109510).

REFERENCES:

Guy, R. K. Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.

Sloane, N. J. A. Sequences A000079/M1129, A000396/M4186, A005100/M0514, A005101/M4825, A028982, A033880, and A109510 in "The On-Line Encyclopedia of Integer Sequences."