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A329357
Numbers whose reversed binary expansion has co-Lyndon factorization of length 2.
3
2, 3, 5, 9, 11, 17, 19, 23, 33, 35, 37, 39, 43, 47, 65, 67, 69, 71, 75, 79, 83, 87, 95, 129, 131, 133, 135, 137, 139, 143, 147, 149, 151, 155, 159, 163, 167, 171, 175, 183, 191, 257, 259, 261, 263, 265, 267, 271, 275, 277, 279, 283, 287, 291, 293, 295, 299
OFFSET
1,1
COMMENTS
First differs from A329327 in lacking 77 and having 83.
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).
EXAMPLE
The reversed binary expansion of each term together with their co-Lyndon factorizations:
2: (01) = (0)(1)
3: (11) = (1)(1)
5: (101) = (10)(1)
9: (1001) = (100)(1)
11: (1101) = (110)(1)
17: (10001) = (1000)(1)
19: (11001) = (1100)(1)
23: (11101) = (1110)(1)
33: (100001) = (10000)(1)
35: (110001) = (11000)(1)
37: (101001) = (10100)(1)
39: (111001) = (11100)(1)
43: (110101) = (11010)(1)
47: (111101) = (11110)(1)
65: (1000001) = (100000)(1)
67: (1100001) = (110000)(1)
69: (1010001) = (101000)(1)
71: (1110001) = (111000)(1)
75: (1101001) = (110100)(1)
79: (1111001) = (111100)(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, #]]&]]];
Select[Range[100], Length[colynfac[Reverse[IntegerDigits[#, 2]]]]==2&]
CROSSREFS
Positions of 2's in A329326.
Binary co-Lyndon words are counted by A001037 and ranked by A329318.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Sequence in context: A157604 A238657 A329327 * A101737 A176550 A364059
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 12 2019
STATUS
approved