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A253953 Numbers that require three steps to collapse to a single digit in base 4 (written in base 4). 3
223, 1213, 2023, 2122, 2203, 2221, 3133, 11113, 12103, 13033, 20023, 20203, 20221, 21202, 22003, 22021, 22201, 22333, 30313, 31033, 31132, 103033, 110113, 111103, 113032, 121003, 200023, 200203, 200221, 202003, 202021 (list; graph; refs; listen; history; text; internal format)



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.


Steve Butler, Table of n, a(n) for n = 1..637

S. Butler, R. Graham and R. Stong, Partition and sum is fast, arXiv:1501.04067 [math.HO], 2014.


As an example a(1)=223 (in base 4).  There are then three ways to insert plus signs in the first step:

2+23   22+3   2+2+3

This gives the numbers (in base 4) as 31, 31, and 13 respectively.  In the second step we have one of the following two:

3+1   1+3

In both cases this gives the number (in base 4) of 10.  Finally in the third step we have the following:


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.)


Cf. A253057, A253058, A253952.

Sequence in context: A094459 A108819 A158226 * A205273 A205266 A152834

Adjacent sequences:  A253950 A253951 A253952 * A253954 A253955 A253956




Steve Butler, Jan 20 2015



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Last modified April 8 01:45 EDT 2020. Contains 333312 sequences. (Running on oeis4.)