[seqfan] An idea for a simple mechanical computer to illustrate two OEIS-sequences.

[Antti Karttunen]

I went to pick up some berries, but came back with this idea instead
(well, in any case, involving small round things):

We have a machine, with several glass tubes, that are labeled as
"2", "3", "5", "7", "11", "13", etc. When machine is in
operation, each tube contains zero or more steel balls that just
fit inside them.  The tubes are open from the top, while at the bottom
of each one there is a mechanical "valve", normally closed, but which
can be electrically opened just for the time it takes for exactly one
ball to come out of the tube (or none if the tube is empty, but never
letting out more than one).  Moreover, at the top of the glass tubes is a
moving "ball-dispenser" that may automatically drop one or more balls to some
of the glass tubes, by the program's instructions.  (I'm partly
inspired by this old Lotto-machine https://youtu.be/_5Dn7mm2WkE?t=21
but of course in our version the tubes must be tall enough to
accommodate several balls, balls that however, need not be specially
marked, but can be totally indistinguishable from each other).

Also, under each bottom-valve of each tube is either an optical or
mechanical sensor that detects a dropped ball, before it is transferred
back up to the dispenser's reservoir of extra balls by some mechanism
hidden to us.

The user activates the machine by manually dropping several balls into
the tubes (>= 0 in each), after which he presses the start button,
from which onward the machine's mechanical program proceeds as follows: the
tubes are scanned from the left to the right, starting, from the tube
labeled "2", with each tube's bottom valve opened in turn to allow a
single ball to come out of the tube, provided there are any balls in that tube.
The dropped ball then passes the above-mentioned "ball-sensor", which
activates a hard-coded "dispensing routine" specific for each tube.

For the tube "2" nothing is done, for the tube "3", for any ball that
comes from the bottom of it, the ball-dispenser drops an extra ball
into the tube "2" (*), for the tube "5", if a ball comes out, then the
ball-dispenser drops two balls into the tube "2", for the tube "7",
for a single ball coming out, the dispenser drops one ball to "2" and one
ball to "3", for "11" one ball to "2" and one to "5", for "13" two
extra balls to "2" and one to "3", and so on.  Each ball-sensor,
when activated by a dropped ball, also rotates an arrow in a clock-like
display, which keeps the total count how many balls goes through the machine.
Additionally, there is another arrow that is rotated by one step
whenever all the tubes have been scanned in turn and the process
returns back to the tube "2". The output of the program are those two
counts, which are ready when all the tubes are finally empty.

(* Here it might be the same ball, if there were a direct intermediate
tube or channel
from the bottom of "3" to the top of "2". For other tubes no such
shortcut-optimizations
are possible, and the dispenser needs to be used.)

If we consider the number of balls in the tubes "2", "3", "5", ..., etc,
as giving the exponents in the prime factorization of n, which two
sequences a(n) and b(n) the counts given by those two arrows correspond to?
(Both are in OEIS).

I don't know whether this kind of machine would make any sense in a
science fair (the mathematics doesn't involve Fibonaccis, but just basic
number theory, no rocket science), but it might make a nice clanking
contraption for Bridges kind of Art festival? Or let the bright kid to
play with it and figure out what's going on? Like e.g., why the machine
always stops, even though for a moment the number of balls often increases?

Also, I wonder if there there other number theoretical iterations or
problems that would allow such simple-minded mechanical
implementations?


Best wishes,

Antti Karttunen