William asked me to forward this to sage-devel:
> the SAGE ECM interface found a first factor of the aliquot sequence starting
> by 552:
>
> remains
> 23648161798622140141259448258749760352819524456141488104537419990481892694930432002158957619604181055633215274583954462907657503167424176909
> (140 digits)
> found factor by ecm: 58417195751812372006463994075468288063413 with
> parameters {'poly': 'Dickson(6)', 'sigma': '300411371', 'B1': '3990569',
> 'B2': '8561602150'}
>
> Other nice factors will surely follow.
> Paul
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.
Paul Zimmermann
PS: for more details, see http://www.loria.fr/~zimmerma/records/aliquot.html
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