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A253952 Numbers that require three steps to collapse to a single digit in base 4 (written in base 10).

%I A253952 #16 Mar 14 2016 09:07:06
%S A253952 43,103,139,154,163,169,223,343,403,463,523,547,553,610,643,649,673,
%T A253952 703,823,847,862,1231,1303,1363,1486,1603,2059,2083,2089,2179,2185,
%U A253952 2209,2239,2434,2563,2569,2593,2623,2689,2731
%N A253952 Numbers that require three steps to collapse to a single digit in base 4 (written in base 10).
%C A253952 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 A253952 Steve Butler, <a href="/A253952/b253952.txt">Table of n, a(n) for n = 1..638</a>
%H A253952 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 A253952 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 A253952 2+23   22+3   2+2+3
%e A253952 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 A253952 3+1   1+3
%e A253952 In both cases this gives the number (in base 4) of 10.  Finally in the third step we have the following:
%e A253952 1+0
%e A253952 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 A253952 Cf. A253057, A253058, A253953.
%K A253952 nonn,base
%O A253952 1,1
%A A253952 _Steve Butler_, Jan 20 2015