[seqfan] Re: 32/25 - a Northern Summer puzzle

[Peter Munn]

On Mon, June 27, 2022 3:47 pm, Peter Munn wrote:
[...]
> How many operations, multiplying or dividing by an integer, does it take
> to get from 1 to 32/25?
[...]
> At each stage, we can find the length of the new shortest route, 2k, and
> the lowest upper bound that permits a route of 2k, and forbid the next
> stage to go as high. So how does the sequence 2, 4, 6, ... (the values of
> 2k) continue? I am confident it is infinite, which means there is an
> asymptote for the upper bound of the restriction range. What is this
> asymptote?

When published, you will be able to find my answer for the asymptote in a
comment dated "Jun 29 2022" in a related sequence: it is the upper bound
of "the critical interval" that I mention.

For my analysis, I have 2 visual aids, the simpler of which is an
arrangement of the 5-smooth rationals, q, 1 <= q < 6 on an infinite square
grid as sampled below:

.  |  243/200  |   81/80  <->  81/16   |  135/32   |  225/64
.  |  243/100  |   81/40   |   27/16   |   45/32   |   75/64
.  |  243/50   |   81/20   |   27/8    |   45/16   |   75/32
.  |   81/50   |   27/20   |    9/8   <->  45/8    |   75/16
.  |   81/25   |   27/10   |    9/4    |   15/8    |   25/16
.  |   27/25  <->  27/5    |    9/2    |   15/4    |   25/8
.  |   54/25   |    9/5    |    3/2    |    5/4    |   25/24
.  |  108/25   |   18/5    |    3/1    |    5/2    |   25/12
.  |   36/25   |    6/5    |    1/1   <->   5/1    |   25/6
.  |   72/25   |   12/5    |    2/1    |    5/3    |   25/18
. <-> 144/25   |   24/5    |    4/1    |   10/3    |   25/9
.  |   48/25   |    8/5    |    4/3    |   10/9   <->  50/9
.  |   96/25   |   16/5    |    8/3    |   20/9    |   50/27
.  |   32/25   |   16/15  <->  16/3    |   40/9    |  100/27
.  |   64/25   |   32/15   |   16/9    |   40/27   |  100/81
. <-> 128/25   |   64/15   |   32/9    |   80/27   |  200/81
.  |  128/75   |   64/45   |   32/27  <-> 160/27   |  400/81

The rational 2^i*3^j*5^k is in the cell in row (i-j) column k. (This is a
1-to-1 mapping from the qualifying rationals onto the full grid.) To plan
a route from 1/1 to q, 1 <= q < 6, note that there are no barriers to
passing up and down the columns, but passing between columns is only
possible at "gateways", indicated "<->". This is where the cell to the
left contains a value less than 6/5 and the cell to the right a value at
least 5, in which case one value is 5 times the other.

If we constrain the route with an upper limit, U < 6, meaning only cells
containing values less than U are available for routes between cells,
passage along any single column becomes limited and the unavailable cells
are more concentrated at gateways. When U = 5, the remaining available
cells are in isolated blocks of 4, 7 or 12 adjacent cells within a column.

With U = 5.5 there are cells in isolated blocks (of 12), but they are much
less dense on the grid than cells that have routes to infinitely many
other cells. Broadly speaking, these long routes run between gateway
cells, angled steeply from top left to bottom right across the grid, and
they corresponding to multiplication, in steps, by 81/80 (most often)
interspersed with 243/256. (Note the gateway cells contain values qg, 5 <=
qg < 5.5.) On the other hand, significant passage orthogonal to these
routes is blocked by what might be viewed as an infinite array of
essentially parallel infinite fences. (This is because the cells of the
grid correspond to an approximate slice through the full 3-dimensional
i,j,k-indexed grid onto which all rational 2^i*3^j*5^k might be mapped;
and the intersection of this slice with the 2-dimensional set of cells
generated by vectors 81/80 and 243/256 from a starting cell is essentially
1-dimensional.)

As U is increased, at a critical value U_crit, the remaining isolated
blocks become unisolated. Moreover, at U = U_crit, if the cell containing
the U_crit value itself was also made available, a single hole would
appear in one of the "fences". I believe any further increase in U creates
holes in all the fences, and then there are routes between any two
5-smooth rationals q_1 and q_2, 1 <= q_1 < q_2 < U. This corresponds to
the newly available gateway cells having new multistep multiplication
routes to the previously available gateways (that is, not a combination of
81/80 and 243/256). Unsurprisingly, the average distance between the holes
in a fence is greater the smaller the value of U - U_crit.

Anyway, I believe the asymptote related to the route from 1 to 32/25 will
be this general U_crit value, and with the mass appearance of "holes in
fences", routes open to very many other rationals. I chose to use 32/25
because of its small product of numerator and denominator.

The sequence starts 2, 4, 6, 12, 26, 76, ... including the terms found by
wnmyers, and the next term is 126.

Best Regards,

Peter Munn