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A216730 List of "rotten" strings in {2,3}* (in the curling number sense). 10
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 (list; graph; refs; listen; history; text; internal format)



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.


Benjamin Chaffin, Table of n, a(n) for n = 1..2400

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102, Dec 25 2012.

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.

Index entries for sequences related to curling numbers


Cf. A094004, A160766, A216950.

Sequence in context: A048376 A053422 A000461 * A048795 A068186 A021284

Adjacent sequences:  A216727 A216728 A216729 * A216731 A216732 A216733




Benjamin Chaffin and N. J. A. Sloane, Sep 16 2012



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Last modified October 17 05:19 EDT 2017. Contains 293467 sequences.