# positive multiple of a semiperfect number is also semiperfect

Just as the theorem on multiples of abundant numbers shows that multiples^{} of abundant numbers are also abundant, it is also true that multiples of semiperfect numbers are also semiperfect, and T. Foregger’s proof of the abundant number theorem lays bare a simple mechanism that we can also employ for semiperfect numbers.

Given the divisors^{} ${d}_{1},\mathrm{\dots},{d}_{k-1}$ of $n$ (where $k=\tau (n)$ and $\tau (x)$ is the divisor function^{}), sorted in ascending order for our convenience, and with a smart iterator $i$ that somehow knows to skip over those divisors that contribute to $n$’s abundance, we can show that the divisors of $nm$ (with $m>0$) will include ${d}_{1}m,\mathrm{\dots},{d}_{k-1}m$. With our smart iterator $i$ and thanks to the distributive property of multiplication, it follows that

$$\sum _{i=1}^{k-1}{d}_{i}m=nm,$$ |

our desired result.

Title | positive multiple of a semiperfect number is also semiperfect |
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Canonical name | PositiveMultipleOfASemiperfectNumberIsAlsoSemiperfect |

Date of creation | 2013-03-22 16:18:57 |

Last modified on | 2013-03-22 16:18:57 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Derivation |

Classification | msc 11A05 |