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 A329395 #9 Jun 09 2020 22:13:48
%S A329395 1,2,3,4,7,8,10,13,15,16,22,25,31,32,36,42,46,49,53,59,63,64,76,82,94,
%T A329395 97,109,115,127,128,136,148,156,162,166,169,170,172,181,182,190,193,
%U A329395 201,202,211,213,214,217,221,227,235,247,255,256,280,292,306,308
%N A329395 Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.
%C A329395 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 A329395 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).
%C A329395 Conjecture: also numbers k such that the k-th composition in standard order (A066099) is a palindrome, cf. A025065, A242414, A317085, A317086, A317087, A335373. - _Gus Wiseman_, Jun 06 2020
%e A329395 The sequence of terms together with their trimmed binary expansions and their co-Lyndon and Lyndon factorizations begins:
%e A329395 1: () = 0 = 0
%e A329395 2: (0) = (0) = (0)
%e A329395 3: (1) = (1) = (1)
%e A329395 4: (00) = (0)(0) = (0)(0)
%e A329395 7: (11) = (1)(1) = (1)(1)
%e A329395 8: (000) = (0)(0)(0) = (0)(0)(0)
%e A329395 10: (010) = (0)(10) = (01)(0)
%e A329395 13: (101) = (10)(1) = (1)(01)
%e A329395 15: (111) = (1)(1)(1) = (1)(1)(1)
%e A329395 16: (0000) = (0)(0)(0)(0) = (0)(0)(0)(0)
%e A329395 22: (0110) = (0)(110) = (011)(0)
%e A329395 25: (1001) = (100)(1) = (1)(001)
%e A329395 31: (1111) = (1)(1)(1)(1) = (1)(1)(1)(1)
%e A329395 32: (00000) = (0)(0)(0)(0)(0) = (0)(0)(0)(0)(0)
%e A329395 36: (00100) = (0)(0)(100) = (001)(0)(0)
%e A329395 42: (01010) = (0)(10)(10) = (01)(01)(0)
%e A329395 46: (01110) = (0)(1110) = (0111)(0)
%e A329395 49: (10001) = (1000)(1) = (1)(0001)
%e A329395 53: (10101) = (10)(10)(1) = (1)(01)(01)
%e A329395 59: (11011) = (110)(1)(1) = (1)(1)(011)
%e A329395 63: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)
%t A329395 lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
%t A329395 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 A329395 colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
%t A329395 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 A329395 Select[Range[100],Length[lynfac[Rest[IntegerDigits[#,2]]]]==Length[colynfac[Rest[IntegerDigits[#,2]]]]&]
%Y A329395 Lyndon and co-Lyndon compositions are (both) counted by A059966.
%Y A329395 Numbers whose reversed binary expansion is Lyndon are A328596.
%Y A329395 Numbers whose binary expansion is co-Lyndon are A275692.
%Y A329395 Cf. A001037, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329394, A329398.
%Y A329395 Cf. A329358, A329399, A329400, A329401.
%K A329395 nonn
%O A329395 1,2
%A A329395 _Gus Wiseman_, Nov 13 2019