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(a)aero.org
