# multiply perfect

Recall that a perfect number is an integer that is the sum
of its aliquot divisors, that is, all of its positive
divisors except itself. Another way to say this is:
*n* is perfect if the sum of all of its positive
divisors, denoted sigma(*n*), is twice *n*. Any
positive integer *n* which divides the sum of its
positive divisors is called **multiply perfect** or
** k-perfect** where

*k*is the

**index**sigma(

*n*)/

*n*. For example, here are the smallest multiply perfect numbers for their index:

index | smallest | name | found by |
---|---|---|---|

2 | 6 | perfect | (ancient) |

3 | 120 | 3-perfect | (ancient) |

4 | 30240 | 4-perfect | Descartes, c. 1638 |

5 | 14182439040 | 5-perfect | Descartes, c. 1638 |

6 | 154345556085770649600 | 6-perfect | Carmichael, 1907 |

Fermat (not Carmichael) was the first to find a 6-perfect number (in 1643):

34111227434420791224041472000.You might want to try your hand at proving the following theorems:

- If
nis 3-perfect and 3 does not dividen, then 3nis 4-perfect.- If
nis 5-perfect and 5 does not dividen, then 5nis 6-perfect.- If general, suppose
pis prime. Ifnisp-perfect andpdoes not dividen, thenpnis (p+1)-perfect.- If 3
nis 4k-perfect and 3 does not dividen, thennis 3k-perfect.

See Also:SigmaFunction, PerfectNumber

Related pages(outside of this work)

- Multiply perfect numbers by Achim Flammenkamp
- Multiply perfect numbers from Wikipedia
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