# 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:

indexsmallestnamefound by
26perfect(ancient)
31203-perfect(ancient)
430240 4-perfectDescartes, c. 1638
5141824390405-perfectDescartes, c. 1638
6 1543455560857706496006-perfectCarmichael, 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 3n is 4-perfect.
• If n is 5-perfect and 5 does not divide n, then 5n 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 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.