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A253952
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Numbers that require three steps to collapse to a single digit in base 4 (written in base 10).
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3
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43, 103, 139, 154, 163, 169, 223, 343, 403, 463, 523, 547, 553, 610, 643, 649, 673, 703, 823, 847, 862, 1231, 1303, 1363, 1486, 1603, 2059, 2083, 2089, 2179, 2185, 2209, 2239, 2434, 2563, 2569, 2593, 2623, 2689, 2731
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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)=43 which in base 4 can be written as 223. 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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