%N Numbers that require three steps to collapse to a single digit in base 4 (written in base 10).
%C One step consists of taking the number in base 4 and inserting some plus signs between the digits with no restrictions and adding the resulting numbers together in base 4. The numbers given here cannot be taken to a single digit in one or two steps. It is known that three steps always suffice to get to a single digit, and that there are infinitely many numbers that require three steps.
%H Steve Butler, <a href="/A253952/b253952.txt">Table of n, a(n) for n = 1..638</a>
%H S. Butler, R. Graham and R. Stong, <a href="http://arxiv.org/abs/1501.04067">Partition and sum is fast</a>, arXiv:1501.04067 [math.HO], 2014.
%e As an example a(1)=43 which in base 4 can be written as 223. There are then three ways to insert plus signs in the first step:
%e 2+23 22+3 2+2+3
%e This gives the numbers (in base 4) as 31, 31, and 13 respectively. In the second step we have one of the following two:
%e 3+1 1+3
%e In both cases this gives the number (in base 4) of 10. Finally in the third step we have the following:
%e Which gives 1, a single digit, and we cannot get to a single digit in one or two steps. (Note, the single digit that we reduce to is independent of the sequence of steps taken.)
%Y Cf. A253057, A253058, A253953.
%A _Steve Butler_, Jan 20 2015