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:
|4||30240||4-perfect||Descartes, c. 1638|
|5||14182439040||5-perfect||Descartes, c. 1638|
Fermat (not Carmichael) was the first to find a 6-perfect number (in 1643):
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.
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