# deficiency

Given an integer $n$ with divisors^{} ${d}_{1},\mathrm{\dots},{d}_{k}$ (where the divisors are in ascending order and ${d}_{1}=1$, ${d}_{k}=n$) the difference

$$2n-\left(\sum _{i=1}^{k}{d}_{i}\right)$$ |

is the deficiency^{} of $n$. Or if one prefers,

$$n-\left(\sum _{i=1}^{k-1}{d}_{i}\right).$$ |

The deficiency is essentially the same thing as the abundance multiplied by $-1$. Thus, the deficiency is positive for deficient numbers, 0 for perfect numbers and negative for abundant numbers.

For example, the divisors of 13 add up to 14, which is 12 less than 26. Therefore, 12 has an deficiency of 12. Another example: the divisors of 14 add up to 24, which is 4 less than 28. The deficiency of the first 72 integers is listed in A033879 of Sloane’s OEIS.

Title | deficiency |
---|---|

Canonical name | Deficiency |

Date of creation | 2013-03-22 16:46:50 |

Last modified on | 2013-03-22 16:46:50 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A05 |

Related topic | Abundance |