

A339608


Numbers whose bijective base2 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 ChenFoxLyndon 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 base2.


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(n1), 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 base2 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 base2 it becomes "11", which is equal to its single nontrivial rotation.
108 is not in this sequence, since in bijective base2 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



