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A253953
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Numbers that require three steps to collapse to a single digit in base 4 (written in base 4).
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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:
1+0
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.)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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