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

(1) |

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

(2) |

(3) |

Define the density function

(4) |

(5) |

(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).

**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.

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1999-05-25