OFFSET
1,2
COMMENTS
First differs from A211100 at a(77) = 3, A211100(77) = 2. The reversed binary expansion of 77 is (1011001), with co-Lyndon factorization (10)(1100)(1), while the binary expansion is (1001101), with Lyndon factorization of (1)(001101).
The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon 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 co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon 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 co-Lyndon words, and if these factors are arranged in certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).
EXAMPLE
The reversed binary expansion of each positive integer together with their co-Lyndon 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 co-Lyndon are A275692.
Length of the co-Lyndon factorization of the binary expansion is A329312.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 11 2019
STATUS
approved