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A216730
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List of "rotten" strings in {2,3}* (in the curling number sense).
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10
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22, 333, 32323, 323232, 2323232, 3232323, 22322232, 23222322, 23223223, 33233233, 223222322, 223222323, 232223222, 332332332, 2232223222, 2232223223, 2232223232, 2322232223, 2322322322, 2332332332, 3322332233, 3323323323, 22322232223, 22322232232, 22322232322, 22322322232, 22322322322, 22323222322, 23222322232, 23223223223
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OFFSET
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1,1
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COMMENTS
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The "curling number" k = k(S) of a string of numbers S = s(1), ..., s(m) is defined as follows. Write S as XY^k for strings X and Y (where Y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of S.
The "tail length" t(S) of S is defined as follows: start with S and repeatedly append the curling number (recomputing it at each step) until a 1 is reached; t(S) is the number of terms that are appended to S before a 1 is reached.
If a 1 is never reached, set t(S)=oo (the Curling Number Conjecture says this will never happen).
A sequence S in {2,3}* is called "rotten" if either of t(2S) or t(3S) (or both) is strictly less than t(S).
Example: S = 32323 has curling number k=2, so we get 323232; now k=3, so we get 3232323; now k=3, so we get 32323233; now k=2, so we get 323232332; now k=1 so we stop. We added 4 terms before reaching 1, so t(S)=4.
On the other hand, 2S = 232323 only extends to 232323321..., so t(2S)=2 which means S is rotten.
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LINKS
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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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