Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3 |

Intel Processor Architecture

2111 NE 25th Avenue

Hillsboro, OR 97124

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John P. Linderman

1028 Prospect Street

Westfield, NJ 07090

USA

N. J. A. Sloane

The OEIS Foundation Inc.

11 South Adelaide Avenue

Highland Park, NJ 08904

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Allan R. Wilks

425 Ridgeview Avenue

Scotch Plains, NJ 07076

USA

**Abstract:**

Given a finite nonempty sequence *S* of integers, write it as *X**Y*^{k}, where
*Y*^{k} is a power of greatest exponent that is a suffix of *S*: this *k* is
the curling number of *S*. The curling number conjecture is that if one
starts with any initial sequence *S*, and extends it by repeatedly
appending the curling number of the current sequence, the sequence will
eventually reach 1. The conjecture remains open. In this paper we
discuss the special case when *S* consists just of 2s and 3s. Even this
case remains open, but we determine how far a sequence consisting of *n*
2s and 3s can extend before reaching a 1, conjecturally for
*n* ≤ 80. We
investigate several related combinatorial problems, such as finding
*c*(*n*, *k*), the number of binary sequences of length *n* and curling number
*k*, and *t*(*n*,*i*), the number of sequences of length *n* which extend for *i*
steps before reaching a 1. A number of interesting combinatorial
problems remain unsolved.

(Concerned with sequences A027375 A090822 A122536 A135491 A160766 A216730 A216813 A216950 A216951 A216955 A217208 A217209 A217437 A217943 A218869 A218870 A218875 A218876.)

Received December 25 2012;
revised version received March 12 2013.
Published in *Journal of Integer Sequences*, March 16 2013.

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