

A329326


Length of the coLyndon factorization of the reversed binary expansion of n.


24



1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 3, 3, 2, 6, 3, 4, 2, 5, 4, 3, 2
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OFFSET

1,2


COMMENTS

First differs from A211100 at a(77) = 3, A211100(77) = 2. The reversed binary expansion of 77 is (1011001), with coLyndon factorization (10)(1100)(1), while the binary expansion is (1001101), with Lyndon factorization of (1)(001101).
The coLyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the coLyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A coLyndon word is a finite sequence that is prime with respect to the coLyndon product. Equivalently, a coLyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into coLyndon words, and if these factors are arranged in certain order, their concatenation is equal to their coLyndon product. For example, (1001) has sorted coLyndon factorization (1)(100).


LINKS

Table of n, a(n) for n=1..87.


EXAMPLE

The reversed binary expansion of each positive integer together with their coLyndon factorizations begins:
1: (1) = (1)
2: (01) = (0)(1)
3: (11) = (1)(1)
4: (001) = (0)(0)(1)
5: (101) = (10)(1)
6: (011) = (0)(1)(1)
7: (111) = (1)(1)(1)
8: (0001) = (0)(0)(0)(1)
9: (1001) = (100)(1)
10: (0101) = (0)(10)(1)
11: (1101) = (110)(1)
12: (0011) = (0)(0)(1)(1)
13: (1011) = (10)(1)(1)
14: (0111) = (0)(1)(1)(1)
15: (1111) = (1)(1)(1)(1)
16: (00001) = (0)(0)(0)(0)(1)
17: (10001) = (1000)(1)
18: (01001) = (0)(100)(1)
19: (11001) = (1100)(1)
20: (00101) = (0)(0)(10)(1)


MATHEMATICA

colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Table[Length[colynfac[Reverse[IntegerDigits[n, 2]]]], {n, 100}]


CROSSREFS

The non"co" version is A211100.
Positions of 2's are A329357.
Numbers whose binary expansion is coLyndon are A275692.
Length of the coLyndon factorization of the binary expansion is A329312.
Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A328596, A329131, A329314, A329318, A329324, A329325.
Sequence in context: A185166 A276555 A211100 * A105264 A063787 A307092
Adjacent sequences: A329323 A329324 A329325 * A329327 A329328 A329329


KEYWORD

nonn


AUTHOR

Gus Wiseman, Nov 11 2019


STATUS

approved



