I have made a model of the Lehmer chain Machine. This has 7 chains of lengths: 10, 11, 13, 16, 17, 19, and 23 logical links. For mechanical reasons I use a pair of links for each position, so each chain has an even number of real links. There are ten teeth on each sprocket, for five logical steps for each revolution of the shaft. The large thing at the left end of the shaft is a ball bearing, which is not being used except to keep the shaft from sliding off the end. Another sprocket at the right serves the same purpose, and provides a handle to turn the shaft.
I have software, written in Assembly and runs at a DOS prompt, to simulate chain machines with chains of arbitrary lengths. A .ZIP file has the Simulation software. The documentation is weak unless you are familiar with DOS.
I have ordered 12 feet of chain with one side tab for each inch of chain. Initially, I though I would make the same chain lengths as in the photo, but after some experimenting I think I will replace the 10 link chain with one of 21 links. With this I will have "representation" for the prime numbers: 2 3 7 11 13 16 17 19 and 23. This change picks up 3, and 7 but, will miss 5.
I have now: (Thu 02-12-12) got the new chain. But, have not cut it up into the various lengths. At the moment I think I will make chains of: 11, 13, 17, 19, 20, 21, and 23 links.
I will probably not finish this until early March. It is not obvious what chain length are best. If any mathematically oriented readers can shed some light on "optional" chain lengths, I would like some advice.