# 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
*n*is 3-perfect and 3 does not divide*n*, then 3*n*is 4-perfect. - If
*n*is 5-perfect and 5 does not divide*n*, then 5*n*is 6-perfect. - If general, suppose
*p*is prime. If*n*is*p*-perfect and*p*does not divide*n*, then*pn*is (*p*+1)-perfect. - If 3
*n*is 4*k*-perfect and 3 does not divide*n*, then*n*is 3*k*-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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