[seqfan] 32/25 - a Northern Summer puzzle

[Peter Munn]

Amongst the data for a little research I have been doing, I saw the
potential for a little puzzle.

How many operations, multiplying or dividing by an integer, does it take
to get from 1 to 32/25? Clearly 2, but what if we restrict the range of
the intermediate values? Let's keep the lower bound as 1. If we are not
allowed as high as 32, we can do it in 4 steps: 1 -> 8 -> 8/5 -> 32/5 ->
32/25. If we are not allowed as high as 8, we can do it in 6 (simple
puzzle, to work out the steps).

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?

Best Regards,

Peter Munn