First of all, your measures of loop size are off by 1 from the usual notation in this kind of problem; e.g., a loop a -> b -> c -> a is considered a 3 loop, not a 2 loop. A fixed point is then a 1 loop. Every chain of this sort will end in a loop of some sort. Consider a number N starting with n+2 8 digits, followed by n 0's, with n >= 1. Any number less than this which transforms to something larger will still transform to number starting with n+2 8's, and then the next term(s) will be smaller until a number less than N is reached. The chain thus cannot grow to infinity, so it must eventually loop. Certainly the length of a chain increases without limit: start with a large enough sequence of digits with the same parity (not 0's or 9's), and you will get a large number of steps in the same direction. The interesting question is whether there are arbitrarily large loops. My guess is that there are not. Essentially the same argument applies to adding even digits and subtracting odd; just look at 9's instead of 8's. Franklin T. Adams-Watters P.S. the list of fixed points should start with 0. That applies to A036301, too - which, incidently, should have the "base" keyword. -----Original Message----- From: Eric.Angelini@kntv.be
... Should be ok now ... Best, É. ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.