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A339608 Numbers whose bijective base-2 representation is a Lyndon word. 2
1, 2, 4, 8, 10, 16, 18, 22, 32, 34, 36, 38, 42, 46, 64, 66, 68, 70, 74, 76, 78, 86, 94, 128, 130, 132, 134, 136, 138, 140, 142, 146, 148, 150, 154, 156, 158, 170, 174, 182, 190, 256, 258, 260, 262, 264, 266, 268, 270, 274, 276, 278, 280, 282, 284, 286, 292, 294, 298, 300, 302, 308 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A Lyndon word is a word which is lexicographically smaller than all its nontrivial rotations.

From the Chen-Fox-Lyndon theorem, every word can be written in a unique way as a concatenation of a nonincreasing sequence of Lyndon words. Since each natural number has a unique string representation in bijective bases, it can also be written exactly one way as a concatenation of these numbers in nonincreasing lexicographic order, in bijective base-2.

LINKS

Harald Korneliussen, Table of n, a(n) for n = 1..20000

Wikipedia, Bijective numeration

Wikipedia, Standard factorization of a Lyndon word

FORMULA

Observation: a(n) = 2*A326774(n-1), n >= 2. (At least for the terms from the Data section). - Omar E. Pol, Dec 09 2020

A007931(a(n)) = A102659(n). - Alois P. Heinz, Dec 09 2020

a(n) = A329327(n) - 1. - Harald Korneliussen, Mar 02 2021

EXAMPLE

1 and 2 are in this sequence, since their bijective base-2 representations are also just "1" and "2", and words of just one letter have no nontrivial rotations.

3 is not in this sequence, since written in bijective base-2 it becomes "11", which is equal to its single nontrivial rotation.

108 is not in this sequence, since in bijective base-2 it becomes "212212", which is larger than two of its nontrivial rotations (both equal to "122122"). However, "212212" can be uniquely split into the lexicographically nonincreasing sequence of Lyndon words "2", "122" and "12", corresponding to 2, 10 and 4 in this sequence.

CROSSREFS

Cf. A007931, A102659, A326774, A329327.

Sequence in context: A125021 A085406 A022340 * A268497 A093886 A125732

Adjacent sequences:  A339605 A339606 A339607 * A339609 A339610 A339611

KEYWORD

nonn,base

AUTHOR

Harald Korneliussen, Dec 09 2020

STATUS

approved

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Last modified June 13 09:02 EDT 2021. Contains 344981 sequences. (Running on oeis4.)