Digital silliness

[hv at crypt.org]

hv at crypt.org wrote:
:"David Wilson" <davidwwilson at comcast.net> wrote:
::For a number n, let f(n) be the set of numbers gotten by splitting n^2 at the 0 
::digits.  For example
::
::29648^2 = 879003904
::
::so f(29648) = { 4, 39, 879 }
::
::Let S be the smallest set of numbers containing 2 and fixed by f.  What is the 
::largest element of S?
:
:Interesting: I wonder whether the set is finite. Clearly it is for
:certain smaller bases: letting a(n) be the largest element of S when
:dealing with base n (so that the question posed asks for a(10)),
:a(2) = a(4) = 2 are obvious; calculating by hand I found a(3) = 5575,
:while a(5) quickly went beyond what was practical by hand.
[...]
:Getting some more results for smaller bases may give a better indication
:of the chances this will terminate. I plan to try converting the program
:to take the base as a parameter.

I've done that, though I don't yet have full confidence in the results
(in part because it isn't finding the same counts for base 10 as the
original program).

It looks like my hand calculation of a(3) was wrong, current results are:
a(2) = 2
a(3) = 1849 ~= 2^11
a(4) = 2
a(5) = 55834574872 ~= 2^36
a(6) = 71936956422890357877389 ~= 2^76
a(7) = 243595882728074946109199629661893196422198513949733 ~= 2^167
a(8) = 4
a(9) > 2^192 (and not necessarily finite, conjecture around 2^770)
a(10) > 2^224 (and not necessarily finite, conjecture around 2^1540)

I'd value some independent confirmation for 5, 6 and 7, or more results.

The current program (C with GMP) took 98 minutes (and 205MB of memory)
to find a(7); I'm pretty sure I can't find a(9) with this approach
unless I trim my memory requirements a lot more, and certainly not a(10).

For the OEIS, I'd also appreciate suggestions for a succinct definition
of the sequence: maybe it would be easier to describe it in terms of
successive sets:
  let S_0 = { 2 }
  let S_{k+1} be the union of f_n(i) for each i in S_k
then a(n) is the greatest element in any S_k. (This may also be useful
for checking whether the results are likely to be finite: the "pend"
counts from my previous message represent successive values of
| S_{k+1} - S{k} |.)

Also of interest would be proof that there exists an n for which a(n) is
_not_ finite, but my hunch at the moment is that a(n) is finite for all n.

The full data for base 3:
S = {1, 2, 4, 7, 8, 13, 16, 22, 23, 25, 26, 43, 49, 214, 233, 238, 484, 1849}
  = {1, 2, 11, 21, 22, 111, 121, 211, 212, 221, 222, 1121, 1211, 21221,
     22122, 22211, 122221, 2112111}_3
S^2 = {1, 11, 121, 1211, 2101, 20021, 100111, 122221, 201121, 212011,
       221001, 2112111, 10021221, 2022211011, 2202110201, 2212200221,
       102220100011, 20102200201021}_3

Hugo