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A253953 Numbers that require three steps to collapse to a single digit in base 4 (written in base 4).

%I A253953 #12 Jan 21 2015 21:57:04
%S A253953 223,1213,2023,2122,2203,2221,3133,11113,12103,13033,20023,20203,
%T A253953 20221,21202,22003,22021,22201,22333,30313,31033,31132,103033,110113,
%U A253953 111103,113032,121003,200023,200203,200221,202003,202021
%N A253953 Numbers that require three steps to collapse to a single digit in base 4 (written in base 4).
%C A253953 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 A253953 Steve Butler, <a href="/A253953/b253953.txt">Table of n, a(n) for n = 1..637</a>
%H A253953 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 A253953 As an example a(1)=223 (in base 4).  There are then three ways to insert plus signs in the first step:
%e A253953 2+23   22+3   2+2+3
%e A253953 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 A253953 3+1   1+3
%e A253953 In both cases this gives the number (in base 4) of 10.  Finally in the third step we have the following:
%e A253953 1+0
%e A253953 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 A253953 Cf. A253057, A253058, A253952.
%K A253953 nonn,base
%O A253953 1,1
%A A253953 _Steve Butler_, Jan 20 2015