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%I #35 Jan 03 2019 13:24:23
%S 1781,3239,3887,11177,14821,33047,41065,43981,98657,131461,393901
%N Complete list of numbers that take three steps to collapse to a single digit in base 3 (written in base 10).
%C From _Simon Demers_, Oct 20 2017: (Start)
%C This an exceptionally nice finite sequence based on a surprisingly simple but nontrivial rule: collapse the number expressed in base 3 by inserting plus signs and adding, while minimizing the number of steps (applications).
%C Butler et al. (2014) proved that any number written in base 2 can be collapsed to a single digit in at most two steps. Any number written in base 3 can be collapsed to a single digit in at most two steps except, surprisingly, for the 11 numbers listed in this sequence. One thing that separates base 3 from larger bases is that there are only 11 base-3 numbers that require three applications!
%C Let m be the sum of the digits in base-3 expansion. Butler et al. (2014) showed that candidates for this sequence must have m < 82.
%C (End)
%C In base 3, the terms are written as 2102222, 11102222, 12022222, 120022222, 202022221, 1200022222, 2002022221, 2020022221, 12000022222, 20200022221, 202000022221. - _Andrey Zabolotskiy_, Oct 20 2017
%H Steve Butler, R. L. Graham, Richard Stong, <a href="http://www.math.ucsd.edu/~ronspubs/mis_17_bases.pdf">Collapsing numbers in bases 2, 3, and beyond</a>, in Gathering for Gardner 10, 2014.
%e In base 3, one of the possible ways to collapse a(1) in three steps is as follows:
%e 2102222 -> 2102 + 222 = 10101 -> 1 + 01 + 01 = 10 -> 1 + 0 = 1.
%Y Cf. A253058, A253952, A253953, A293929.
%K nonn,fini,full,nice
%O 1,1
%A _N. J. A. Sloane_, Jan 13 2015
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