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 A329314 #12 Nov 15 2019 09:35:07
%S A329314 1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,1,3,1,1,1,1,1,1,2,
%T A329314 1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,3,1,1,4,1,2,1,1,1,2,2,1,3,1,1,4,1,1,
%U A329314 1,1,1,1,1,3,1,1,2,1,1,1,3,1,1,1,1,1,1
%N A329314 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n.
%C A329314 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).
%e A329314 Triangle begins:
%e A329314 0: () 20: (1211) 40: (12111) 60: (111111)
%e A329314 1: (1) 21: (122) 41: (123) 61: (11112)
%e A329314 2: (11) 22: (131) 42: (1221) 62: (111111)
%e A329314 3: (11) 23: (14) 43: (15) 63: (111111)
%e A329314 4: (111) 24: (11111) 44: (1311) 64: (1111111)
%e A329314 5: (12) 25: (113) 45: (132) 65: (16)
%e A329314 6: (111) 26: (1121) 46: (141) 66: (151)
%e A329314 7: (111) 27: (113) 47: (15) 67: (16)
%e A329314 8: (1111) 28: (11111) 48: (111111) 68: (1411)
%e A329314 9: (13) 29: (1112) 49: (114) 69: (16)
%e A329314 10: (121) 30: (11111) 50: (1131) 70: (151)
%e A329314 11: (13) 31: (11111) 51: (114) 71: (16)
%e A329314 12: (1111) 32: (111111) 52: (11211) 72: (13111)
%e A329314 13: (112) 33: (15) 53: (1122) 73: (133)
%e A329314 14: (1111) 34: (141) 54: (1131) 74: (151)
%e A329314 15: (1111) 35: (15) 55: (114) 75: (16)
%e A329314 16: (11111) 36: (1311) 56: (111111) 76: (1411)
%e A329314 17: (14) 37: (15) 57: (1113) 77: (16)
%e A329314 18: (131) 38: (141) 58: (11121) 78: (151)
%e A329314 19: (14) 39: (15) 59: (1113) 79: (16)
%t A329314 lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
%t A329314 lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
%t A329314 Table[Length/@lynfac[If[n==0,{},IntegerDigits[n,2]]],{n,0,50}]
%Y A329314 Row lengths are A211100.
%Y A329314 Row sums are A029837, or, if the first term is 1, A070939.
%Y A329314 Ignoring the first digit gives A329325.
%Y A329314 Positions of rows of length 2 are A329327.
%Y A329314 Binary Lyndon words are counted by A001037 and ranked by A102659.
%Y A329314 Numbers whose reversed binary expansion is a Lyndon word are A328596.
%Y A329314 Length of the co-Lyndon factorization of the binary expansion is A329312.
%Y A329314 Cf. A059966, A211097, A275692, A296372, A329313, A329315, A329318.
%K A329314 nonn,tabf
%O A329314 0,10
%A A329314 _Gus Wiseman_, Nov 11 2019