# abundant number

An integer $n$ is an abundant number if the sum of the proper divisors of $n$ is more than $n$ itself, or the sum of all the divisors^{} is more than twice $n$. That is, $\sigma (n)>2n$, with $\sigma (n)$ being the sum of divisors function.

For example, the integer 30. Its proper divisors are 1, 2, 3, 5, 6, 10, 15, which add up to 42.

Multiplying a perfect number by some integer $x$ gives an abundant number (as long as $x>1$).

Given a pair of amicable numbers, the lesser of the two is abundant, its proper divisors adding up to the greater of the two.

Title | abundant number |
---|---|

Canonical name | AbundantNumber |

Date of creation | 2013-03-22 15:52:21 |

Last modified on | 2013-03-22 15:52:21 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 6 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A05 |

Related topic | AmicableNumbers |