All aliquot sequences end in some sort of generalized aliquot cycle. A cycle of length zero is just a prime number:

some n -> p -> 1, and the aliquot function of 1 is not defined (since 1 has no proper divisors).

A cycle of length one is a perfect number:

some n -> P -> P -> P -> ...

A cycle of length two is an amicable pair:

some n -> P -> Q -> P -> Q -> P -> ...

Sociable numbers are those who form a cycle of length 3 or more:

some n -> P -> Q -> R -> P -> Q -> R -> ...

or

some n -> P -> Q -> R -> S -> P -> Q -> R -> S

Any aliquot sequence that doesn't end in some sort of cycle (length 0 or more) must, by necessity, have infinitely many terms (which thus by necessity are arbitrarily large). We don't know if any such sequences exist, and answering that question is the point of this forum.

[edit: crosspost, I was replying to the guy on the former page, idea is that no sequence ends in zero]

You have to sum the proper divisors and 1, to get to the next term. For example if your term is 10, you sum 1+2+5=8, your next term is 8. (10 is not a proper divisor of 10).
No sequence ends in zero, as you at least have to add 1 to the sum. When you get to a prime (or start with a prime), then 1 is the only thing you add, as there are no other proper divisors, and the sequence ends in 1.

All sequences end in 1 or in a sociable chain [edit: what is called above "generalized aliquot cycle"]. That is because I believe the famous conjecture is true. If you believe that the famous conjecture is false, then some sequences go to infinity.

c187 finished a bit earlier than expected:

[CODE]Fri Aug 19 23:59:24 2016 prp83 factor: 81686847473504758006224548165679412739660168673296296131927624175642936458151919111
Fri Aug 19 23:59:24 2016 prp104 factor: 18932362263513586512787823670585462560896906573207638060424099856662477952354307120700802524336003290743[/CODE]

Congrats on losing the first (hardest) 10 digits from the top, Ryan! No more c19X-ies in the near future.

10 done, 190 to go! :rolleyes:

Just trimmed another digit off... :smile:

don't count the fish!

The c188 blocker at index 5358 has thus far resisted ECM, firing off GNFS...

[CODE]linear algebra completed 15687 of 14408801 dimensions (0.1%, ETA 85h53m) [/CODE]

c188 done:

[CODE]Thu Aug 25 17:26:51 2016 p64 factor: 3876216633055061419501730313594001806409661150743884472980900647
Thu Aug 25 17:26:51 2016 p124 factor: 8032181759747525191246997851763484249697240356342526142842271225485394373575011306918512486599998419192876604004337942300257
[/CODE]

Congrats on shedding the next 10 digits!

[SPOILER](the intervals between these congrats will soon be shorter, and shorter, and shorter...
For every ten digits, the gnfs jobs become 4 times faster!
The only remaining ingredient is luck. So, good luck on the ride down!)[/SPOILER]

[QUOTE=Batalov;441282]re:spoiler[/QUOTE]
No congrats under 170 digits. After a C196 gnfs, those are piece of cake for him. So, no congrats for him anymore... :razz:

edit: only one congrats, when it terminates! yeaaaa!

I've decreased the update interval for 4788 on [url]http://www.rechenkraft.net/aliquot/AllSeq.html[/url] to twice a day for those who don't follow it directly on factordb.com.