Perfect Number play

So there is this set of numbers out there that just happen to have their proper divisors add up to itself.  For some reason, everyone wants to call them "Perfect numbers."  I think thats a bit presumptuous, but whatever.  I didn't make it up.  So a perfect number, again, is defined as a number who's divisors add up to itself.  See the wikipedia article and the OEIS list for more info on them.

Well, what if you continued to apply the "perfect number operation" onto a number and then its outcome over and over again?  Random example:


216 has the divisors 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72 and 108.
so
1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 27 + 36 + 54 + 72 + 108 =  384

so then we look at 384 who's divisors are

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128 and 192

so
1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 32 + 48 + 64 + 96 + 128 + 192 = 636

and so on with 636 which turns into 876
876 -> 1196
1196 -> 1156
1156 -> 993
993 -> 335
335 -> 73
73 -> 1

and bingo bango we have come to an end.

Not all numbers come to a small end like this though.  Some of the numbers get HUGE (30 out of the first 1000 actually)  I didn't calculate past 100,000,000 ( 100 million)  - better things to do you see.  Also, some of the numbers get caught into infinite loops.  Once you come across one of the perfect numbers, they are their own outcome and its futile to work on it further.  Also 220's outcome is 284 - and 284's is 220!!  Also 1184's is 1210 and 1210 is 1184!!!  very interesting relations there.

SO I present to you the list from 10 to 1000 of continued perfect number operations.  I need to come up with better names for things....what should I call this?

(seems the service I was using to link my large text files doesn't want me to.  I need to find a different one.  Anyone have any suggestions?)

One of the neat ones:

864 -> 1656 -> 3024 -> 6896 -> 6496 -> 8624 -> 12580 -> 16148 -> 14764 -> 11080 -> 13940 -> 17812 -> 14304 -> 23496 -> 41304 -> 62016 -> 120864 -> 196656 -> 343488 -> 565832 -> 495118 -> 316322 -> 158164 -> 118630 -> 94922 -> 52150 -> 59450 -> 57730 -> 51134 -> 27754 -> 13880 -> 17440 -> 24140 -> 30292 -> 22726 -> 14498 -> 9262 -> 5930 -> 4762 -> 2384 -> 2266 -> 1478 -> 742 -> 554 -> 280 -> 440 -> 640 -> 890 -> 730 -> 602 -> 454 -> 230 -> 202 -> 104 -> 106 -> 56 -> 64 -> 63 -> 41 -> 1