# colossally abundant number

An integer $n$ is a colossally abundant number if there is an exponent $\u03f5>1$ such that the sum of divisors^{} of $n$ divided by $n$ raised to that exponent is greater than or equal to the sum of divisors of any other integer $k>1$ divided by $k$ raised to that same exponent. That is,

$$\frac{\sigma (n)}{{n}^{\u03f5}}\ge \frac{\sigma (k)}{{k}^{\u03f5}},$$ |

with $\sigma (n)$ being the sum of divisors function.

The first few colossally abundant numbers are 1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320. The index of a colossally abundant number is equal to the number of its nondistinct prime factors^{}, that is to say that for the $i$th colossally abundant number ${c}_{i}$ the equality $i=\mathrm{\Omega}({c}_{i})$ is true.

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

Canonical name | ColossallyAbundantNumber |

Date of creation | 2013-03-22 17:37:52 |

Last modified on | 2013-03-22 17:37:52 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 4 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A05 |