This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329358 #7 Nov 15 2019 21:36:43
%S A329358 1,3,5,7,9,15,17,21,27,31,33,45,51,63,65,73,74,83,85,86,89,93,99,107,
%T A329358 119,127,129,138,150,153,163,165,174,177,185,189,195,203,205,219,231,
%U A329358 255,257,266,273,274,278,291,294,297,302,305,310,313,323,325,333,341
%N A329358 Numbers whose binary expansion has Lyndon and co-Lyndon factorizations of equal lengths.
%C A329358 We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
%C A329358 Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
%F A329358 A211100(a(n)) = A329312(a(n)).
%e A329358 The binary expansions of the initial terms together with their Lyndon and co-Lyndon factorizations:
%e A329358 1: (1) = (1) = (1)
%e A329358 3: (11) = (1)(1) = (1)(1)
%e A329358 5: (101) = (1)(01) = (10)(1)
%e A329358 7: (111) = (1)(1)(1) = (1)(1)(1)
%e A329358 9: (1001) = (1)(001) = (100)(1)
%e A329358 15: (1111) = (1)(1)(1)(1) = (1)(1)(1)(1)
%e A329358 17: (10001) = (1)(0001) = (1000)(1)
%e A329358 21: (10101) = (1)(01)(01) = (10)(10)(1)
%e A329358 27: (11011) = (1)(1)(011) = (110)(1)(1)
%e A329358 31: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)
%e A329358 33: (100001) = (1)(00001) = (10000)(1)
%e A329358 45: (101101) = (1)(011)(01) = (10)(110)(1)
%e A329358 51: (110011) = (1)(1)(0011) = (1100)(1)(1)
%e A329358 63: (111111) = (1)(1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)(1)
%e A329358 65: (1000001) = (1)(000001) = (100000)(1)
%e A329358 73: (1001001) = (1)(001)(001) = (100)(100)(1)
%e A329358 74: (1001010) = (1)(00101)(0) = (100)(10)(10)
%e A329358 83: (1010011) = (1)(01)(0011) = (10100)(1)(1)
%t A329358 lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
%t A329358 lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
%t A329358 colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A329358 colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
%t A329358 Select[Range[100],Length[lynfac[IntegerDigits[#,2]]]==Length[colynfac[IntegerDigits[#,2]]]&]
%Y A329358 The version counting compositions is A329394.
%Y A329358 The version ignoring the most significant digit is A329395.
%Y A329358 Binary Lyndon/co-Lyndon words are counted by A001037.
%Y A329358 Lyndon/co-Lyndon compositions are counted by A059966.
%Y A329358 Lyndon compositions whose reverse is not co-Lyndon are A329324.
%Y A329358 Binary Lyndon/co-Lyndon words are constructed by A102659 and A329318.
%Y A329358 Cf. A060223, A211100, A275692, A328596, A329312, A329313, A329326, A329398.
%K A329358 nonn
%O A329358 1,2
%A A329358 _Gus Wiseman_, Nov 15 2019