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.
