The picture graphs what happens to the number 14316 after one iterates a sum-of-aliquot-divisors function 28 times: One encounters the 14316 again, so it is a cycle. The application of the function on different starting numbers is a recreational pastime for many a mathematically-minded computer owner and, as such, I was put on the course of pursuing the fate of the number 46758 one year ago today.

It was not my idea. Richard Guy was stuck on factoring term #776 of the sequence and I decided to see if I could at least get as far on my old machine as did he. I needed 11 months to accomplish this and then another 25.5 days to factor term #776. I thought I'd let Richard know. He informed me that he had already reached term #900 and that a computer-science colleague, Mike Jacobson, had reached term #1000, closing his email response with "We amateurs had better now leave it to the professionals?"

I need a new computer.

Update (16 September 2010): I've just learned of an online factor database that will generate thousands of these terms in under 30 seconds!