20171108, 18:30  #1 
"Andrew Booker"
Mar 2013
1010101_{2} 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^{4}·3·5 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
1010101_{2} 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. 
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