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A329312
Length of the co-Lyndon factorization of the binary expansion of n.
43
1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 2, 3, 2, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3
OFFSET
1,3
COMMENTS
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 a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).
Also the length of the Lyndon factorization of the inverted binary expansion of n, where the inverted digits are 1 minus the binary digits.
EXAMPLE
The binary indices of 1..20 together with their co-Lyndon factorizations are:
1: (1) = (1)
2: (10) = (10)
3: (11) = (1)(1)
4: (100) = (100)
5: (101) = (10)(1)
6: (110) = (110)
7: (111) = (1)(1)(1)
8: (1000) = (1000)
9: (1001) = (100)(1)
10: (1010) = (10)(10)
11: (1011) = (10)(1)(1)
12: (1100) = (1100)
13: (1101) = (110)(1)
14: (1110) = (1110)
15: (1111) = (1)(1)(1)(1)
16: (10000) = (10000)
17: (10001) = (1000)(1)
18: (10010) = (100)(10)
19: (10011) = (100)(1)(1)
20: (10100) = (10100)
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[IntegerDigits[n, 2]]], {n, 100}]
CROSSREFS
The non-"co" version is A211100.
Positions of 1's are A275692.
The reversed version is A329326.
Sequence in context: A136107 A178691 A329313 * A211271 A124768 A321014
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 10 2019
STATUS
approved