20171108, 18:30  #1 
"Andrew Booker"
Mar 2013
5×17 Posts 
Extending an aliquot sequence backwards
The aliquot sequence with start value 461214 is the current longest known open ended sequence starting below 1e6 (it merges with the 4788 sequence around index 6000). Right now it's just shy of 19k iterations. Just for fun, I thought I would try to extend the sequence backwards using Goldbach so as to make a longer record.
It's easy to check that s(461214)=s(670097^2). Given an odd number n > 8, we can (conjecturally) write n=p+q+1=s(pq) for distinct primes p,q. Starting from n=670097^2, I found the smallest p such that q=n1p is a probable prime, replaced n by pq, and repeated for 1000 iterations. The sequence of p values is attached. I then used Primo to certify the primality of the first 700 values of q, the largest of which has 2490 digits. It's getting very slow, but I might let it run up to 1000. I'm aware that none of this serves any real purpose, but if anyone would like to join me in this quest, feel free. 
20171108, 20:01  #2 
Oct 2011
2^{2}·5·19 Posts 
I did the same kind of exercise, see on this page (http://www.aliquotes.com/remonter_suite_envers.html) my go backwards of 3, 7, 11, 13, 17 on more than 2000 iterations, as well as the go backwards of 2005020 on more than 1000 iterations (I was talking about it here #7 : http://www.mersenneforum.org/showthread.php?t=18641)

20171109, 20:48  #3 
"Andrew Booker"
Mar 2013
125_{8} Posts 
Yet more evidence that no idea is original. The only thing I can think to add is the primality certificates. It's reasonable to generate them with Primo for 1000 iterations or so.
Good idea putting the sequence on factordb; I thought it might be too large for that, but I guess not. Will factordb compute the sequence for me if I just enter the 3000digit starting value? Also, I don't fancy uploading 1000 certificates by hand; if anyone has a script that they don't mind sharing, that would be a great help. 
20201026, 13:54  #4 
Nov 2016
2^{2}·691 Posts 
The number 1578 merges to 56440, but it is not known whether the sequence of 56440 (currently at 166 digits) also merges to 56440, thus it is not known whether 56440 is member of sociable number cycle (but if so, then this cycle will contain 6000+ numbers ....)

20201026, 14:05  #5 
"Ed Hall"
Dec 2009
Adirondack Mtns
2×3^{2}×197 Posts 
factordb would pick up a cycle. Since all the factors of all the terms of a sequence exist within the database, it would not be shown as open ended. As to whether this eventually happens, only more factoring will tell. . .

20201107, 15:58  #6 
Nov 2016
2^{2}·691 Posts 
so the sequences A003416 and A122726 might miss numbers (i.e. it is unknown whether there are members of sociable number cycle <= 1799281330 other than numbers listed in A003416), however, the smallest number whose aliquot sequence has not yet been fully determined (276) cannot be member of sociable number cycle, since it is an untouchable number (A005114), however, the number 56440 might be.
Last fiddled with by sweety439 on 20201107 at 16:01 
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