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 A329325 #8 Nov 12 2019 04:20:04
%S A329325 1,1,1,1,2,1,1,1,1,1,1,1,3,2,1,3,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,4,3,1,
%T A329325 4,2,1,1,2,2,3,1,4,1,1,1,1,1,3,1,2,1,1,3,1,1,1,1,1,1,2,1,1,1,1,1,1,1,
%U A329325 1,1,1,1,1,1,5,4,1,5,3,1,1,5,4,1,5,2,1
%N A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.
%C A329325 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. 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 A329325 Triangle begins:
%e A329325 1: () 21: (22) 41: (23) 61: (1112)
%e A329325 2: (1) 22: (31) 42: (221) 62: (11111)
%e A329325 3: (1) 23: (4) 43: (5) 63: (11111)
%e A329325 4: (11) 24: (1111) 44: (311) 64: (111111)
%e A329325 5: (2) 25: (13) 45: (32) 65: (6)
%e A329325 6: (11) 26: (121) 46: (41) 66: (51)
%e A329325 7: (11) 27: (13) 47: (5) 67: (6)
%e A329325 8: (111) 28: (1111) 48: (11111) 68: (411)
%e A329325 9: (3) 29: (112) 49: (14) 69: (6)
%e A329325 10: (21) 30: (1111) 50: (131) 70: (51)
%e A329325 11: (3) 31: (1111) 51: (14) 71: (6)
%e A329325 12: (111) 32: (11111) 52: (1211) 72: (3111)
%e A329325 13: (12) 33: (5) 53: (122) 73: (33)
%e A329325 14: (111) 34: (41) 54: (131) 74: (51)
%e A329325 15: (111) 35: (5) 55: (14) 75: (6)
%e A329325 16: (1111) 36: (311) 56: (11111) 76: (411)
%e A329325 17: (4) 37: (5) 57: (113) 77: (6)
%e A329325 18: (31) 38: (41) 58: (1121) 78: (51)
%e A329325 19: (4) 39: (5) 59: (113) 79: (6)
%e A329325 20: (211) 40: (2111) 60: (11111) 80: (21111)
%e A329325 For example, the trimmed binary expansion of 41 is (01001), with Lyndon factorization (01)(001), so row 41 is {2,3}.
%t A329325 lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
%t A329325 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 A329325 Table[Length/@lynfac[Rest[IntegerDigits[n,2]]],{n,100}]
%Y A329325 Row lengths are A211097.
%Y A329325 Row sums are A000523.
%Y A329325 Keeping the first digit gives A329314.
%Y A329325 Positions of singleton rows are A329327.
%Y A329325 Binary Lyndon words are counted by A001037 and ranked by A102659.
%Y A329325 Numbers whose reversed binary expansion is a Lyndon word are A328596.
%Y A329325 Length of the co-Lyndon factorization of the binary expansion is A329312.
%Y A329325 Cf. A059966, A211097, A257250, A275692, A296372, A328594, A329313, A329315, A329318.
%K A329325 nonn,tabf
%O A329325 1,5
%A A329325 _Gus Wiseman_, Nov 11 2019