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

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

KEYWORD

nonn,base

AUTHOR

Harald Korneliussen, Dec 09 2020

STATUS

approved