%I #9 Jul 02 2021 01:55:22

%S 1,2,3,4,6,7,8,10,12,14,15,16,20,24,26,28,30,31,32,36,38,40,42,48,50,

%T 52,54,56,58,60,62,63,64,72,80,84,96,98,100,104,106,108,112,114,116,

%U 118,120,122,124,126,127,128,136,140,142,144,160,164,168,170,192

%N Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

%C 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).

%C A sequence of words is uniform if they all have the same length.

%e The sequence of terms together with their co-Lyndon factorizations begins:

%e 1: (1) = (1)

%e 2: (10) = (10)

%e 3: (11) = (1)(1)

%e 4: (100) = (100)

%e 6: (110) = (110)

%e 7: (111) = (1)(1)(1)

%e 8: (1000) = (1000)

%e 10: (1010) = (10)(10)

%e 12: (1100) = (1100)

%e 14: (1110) = (1110)

%e 15: (1111) = (1)(1)(1)(1)

%e 16: (10000) = (10000)

%e 20: (10100) = (10100)

%e 24: (11000) = (11000)

%e 26: (11010) = (11010)

%e 28: (11100) = (11100)

%e 30: (11110) = (11110)

%e 31: (11111) = (1)(1)(1)(1)(1)

%e 32: (100000) = (100000)

%e 36: (100100) = (100)(100)

%e 38: (100110) = (100)(110)

%e 40: (101000) = (101000)

%e 42: (101010) = (10)(10)(10)

%t colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];

%t 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 Select[Range[100],SameQ@@Length/@colynfac[IntegerDigits[#,2]]&]

%Y Numbers whose binary expansion has uniform Lyndon factorization are A023758.

%Y Numbers whose reversed binary expansion is Lyndon are A328596.

%Y Numbers whose binary expansion is co-Lyndon are A275692.

%Y Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.

%Y Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329398.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 13 2019