Dan raises a question which has already been anticipated by some funsters. Many of you will now have several definitions. I (&/or Paul Vaderlind) originally meant `chain of order n beginning with n', but others have noted that they could begin with any number > 1. There is a question about `cyclic' chains (necklaces?). These have been defined as those whose first member divides n(n+1)/2, though that seems even more artificial. No chain can be truly cyclic, though one can ask for chains, or even necklaces, in which each member divides the sum of the previous k members. This is not possible for k = 1 or 2 [except for 4 2 3 1 -- which won't quite make a necklace, 'cos 2 doesn't divide 4+1], but perhaps for some larger k ? [for the first k members, `divisor of sum of all previous members'] Example of chain or necklace, or proof of impossibility ? R. On Tue, 4 May 2004, Daniel Asimov wrote:
Richard,
[snip]
Do you mean to define a divisor chain or order n as necessarily beginning with n ???
(That may lead to a trickier class of questions, but it seems a bit artificial as a definition.)
[snip]
Dan Asimov Visiting Scholar Math Dept. Univ. of Calif. Berkeley, California (510) 524-7789