Abundant Number

An abundant number is an Integer $n$ which is not a Perfect Number and for which

$$s(n) \equiv \sigma(n)-n > n,$$ (1)

where $\sigma(n)$ is the Divisor Function. The quantity $\sigma(n)-2n$ is sometimes called the Abundance. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (Sloane's A005101). Abundant numbers are sometimes called Excessive Numbers.

There are only 21 abundant numbers less than 100, and they are all Even. The first Odd abundant number is

$$945 = 3^3 \cdot 7 \cdot 5.$$ (2)

That 945 is abundant can be seen by computing

$$s(945) = 975 > 945.$$ (3)

Any multiple of a Perfect Number or an abundant number is also abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.

Define the density function

$$A(x)\equiv \lim_{n\to\infty}{\vert\{n:\sigma(n)\geq xn\}\vert\over n}$$ (4)

for a Positive Real Number $x$, then Davenport (1933) proved that $A(x)$ exists and is continuous for all $x$, and Erdös (1934) gave a simplified proof (Finch). Wall (1971) and Wall et al. (1977) showed that

$$0.2441< A(2)<0.2909,$$ (5)

and Deléglise showed that

$$0.2474<A(2)<0.2480.$$ (6)

A number which is abundant but for which all its Proper Divisors are Deficient is called a Primitive Abundant Number (Guy 1994, p. 46).

See also Aliquot Sequence, Deficient Number, Highly Abundant Number, Multiamicable Numbers, Perfect Number, Practical Number, Primitive Abundant Number, Weird Number

References

Deléglise, M. ``Encadrement de la densité des nombres abondants.'' Submitted.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3-33, 1952.

Erdös, P. ``On the Density of the Abundant Numbers.'' J. London Math. Soc. 9, 278-282, 1934.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/abund/abund.html

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

Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, pp. 11 and 13, 1997.

Sloane, N. J. A. Sequence A005101/M4825 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Wall, C. R. ``Density Bounds for the Sum of Divisors Function.'' In The Theory of Arithmetic Functions (Ed. A. A. Gioia and D. L. Goldsmith). New York: Springer-Verlag, pp. 283-287, 1971.

Wall, C. R.; Crews, P. L.; and Johnson, D. B. ``Density Bounds for the Sum of Divisors Function.'' Math. Comput. 26, 773-777, 1972.

Wall, C. R.; Crews, P. L.; and Johnson, D. B. ``Density Bounds for the Sum of Divisors Function.'' Math. Comput. 31, 616, 1977.