The SAGE ECM interface found a first factor of the aliquot sequence starting
found factor by ecm: 584171957518123720064639940754
Other nice factors will surely follow.
An aliquot sequence is simply the iteration of the function n -> sigma(n)-n,
where sigma(n) is the "sum of divisors" function. One open question from
Catalan is whether this sequence always converges to 1 (or to a cycle). The
first to perform extensive computations on aliquot sequences was Lehmer, who
found that all sequences starting from n <= 1000 converge, except perhaps
n=276, 552, 564, 660 and 966. These are the "Lehmer five" sequences. Since
several years, together with other people, I try to extend these Lehmer five
sequences. The main difficulty is that to compute sigma(n), you have to
factor n. For the current large numbers we encounter (150-160 digits) we use
a combination of different algorithms (ECM, QS, NFS). I have now converted
to SAGE the script that (tries to) extend aliquot sequences. The above
factorization is a first success of the new script.