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

On Curling Numbers of Integer Sequences

Benjamin Chaffin
Intel Processor Architecture
2111 NE 25th Avenue
Hillsboro, OR 97124

John P. Linderman
1028 Prospect Street
Westfield, NJ 07090

N. J. A. Sloane
The OEIS Foundation Inc.
11 South Adelaide Avenue
Highland Park, NJ 08904

Allan R. Wilks
425 Ridgeview Avenue
Scotch Plains, NJ 07076


Given a finite nonempty sequence S of integers, write it as XYk, where Yk 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.

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(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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