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Abundant Number

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An abundant number, sometimes also called an excessive number, is a positive integer n for which

s(n)=sigma(n)-n>n,
(1)

where sigma(n) is the divisor function and s(n) is the restricted divisor function. The quantity sigma(n)-2n is sometimes called the abundance.

A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number (Guy 1994, p. 46).

The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).

Every positive integer n>6 with (mod n)60 is abundant. Every proper positive multiple of a perfect number is abundant, as is every positive multiple of an abundant number. Prime numbers are not abundant. Every integer greater than 20161 can be expressed as a sum of two abundant numbers.

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

945=3^3·7·5.
(2)

That 945 is abundant can be seen by computing

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

For a positive real number x, define the density function

A(x)=lim_(n->infty)(|{1<=k<=n:sigma(k)>=xk}|)/n,
(4)

where |B| is the cardinal number of a set B (correcting the expression in Finch 2003, p. 126). Davenport (1933) proved that A(x) exists and is continuous for all x>0, and Erdős (1934) gave a simplified proof (Finch 2003). The special case A(2) then gives the natural density of abundant numbers,

A(2)=lim_(n->infty)(# abundant numbers <=n)/n.
(5)

The following table summarizes improvements in bounds on the constant over time.

valuereference
0.241<A(2)<0.314Behrend (1933)
0.2441<A(2)<0.2909Wall (1971) and Wall et al. (1977)
0.2474<A(2)<0.2480Deléglise (1998)
0.2476171<A(2)<0.2476475Kobayashi (2010, p. 12)

See also

Abundance, Aliquot Sequence, Colossally Abundant Number, Deficient Number, Highly Composite Number, Multiamicable Numbers, Perfect Number, Practical Number, Primitive Abundant Number, Weird Number

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References

Behrend, F. "Über numeri abundantes." Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 21/23, 322-328, 1932.Behrend, F. "Über numeri abundantes. II." Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6, 280-293, 1933.Davenport, H. "Über numeri abundantes." Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6, 830-837, 1933.Deléglise, M. "Bounds for the Density of Abundant Integers." Exp. Math. 7, 137-143, 1998.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 3-33, 2005.Erdős, P. "On the Density of the Abundant Numbers." J. London Math. Soc. 9, 278-282, 1934.Finch, S. R. "Abundant Numbers Density Constant." §2.11 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 126-127, 2003.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-46, 1994.Kobayashi, M. "On the Density of Abundant Numbers." PhD thesis. Hanover, NH: Dartmouth College, 2010.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 "The On-Line Encyclopedia of Integer Sequences."Souissi, M. Un texte manuscrit d'Ibn Al-Bannā' Al-Marrakusi sur les nombres parfaits, abondants, deficients, et amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975.Wall, C. R. "Density Bounds for the Sum of Divisors Function." In The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29-May 1, 1971. (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.

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Abundant Number

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Weisstein, Eric W. "Abundant Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbundantNumber.html

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