R.K.Guy wrote (approximately) a division chain for n is the sequence of numbers from 1 to n, arranged so that each term is a divisor of the sum of the preceding ones i wrote a program to compute them so i could look at them here are the counts up to n=27 (more are pending; this is the simplest program i could think of, and it is not very smart) 1 of length 1 1 of length 2 1 of length 3 1 of length 4 2 of length 5 2 of length 6 3 of length 7 6 of length 8 14 of length 9 12 of length 10 15 of length 11 17 of length 12 29 of length 13 24 of length 14 178 of length 15 128 of length 16 156 of length 17 140 of length 18 182 of length 19 507 of length 20 2210 of length 21 1636 of length 22 2272 of length 23 9455 of length 24 17437 of length 25 30202 of length 26 102292 of length 27 there are two curious phenomena i notice by looking at the chains themselves: there are very limited choices for the first and last element (1) first elements seem to be all large values (mostly, but not always, all consecutive values from some point up to the largest) (2) last elements seem to be extremely limited (mostly, but not always, no more than 3 different choices) i'll put the entire file, with all chains, counts, and the program, out on the web when the program finishes (unless somebody disputes the counts, in which case i'll post only the program on the web instead - it is only 160 lines of C code - and we can figure out what i did wrong) more soon, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal@aero.org