A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

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, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 136, 140, 142, 144, 160, 164, 168, 170, 192

Offset: 1

Author

Gus Wiseman, Nov 13 2019

Keywords

• nonn

Comments

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).

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

Examples

			The sequence of terms together with their co-Lyndon factorizations begins:
   1:      (1) = (1)
   2:     (10) = (10)
   3:     (11) = (1)(1)
   4:    (100) = (100)
   6:    (110) = (110)
   7:    (111) = (1)(1)(1)
   8:   (1000) = (1000)
  10:   (1010) = (10)(10)
  12:   (1100) = (1100)
  14:   (1110) = (1110)
  15:   (1111) = (1)(1)(1)(1)
  16:  (10000) = (10000)
  20:  (10100) = (10100)
  24:  (11000) = (11000)
  26:  (11010) = (11010)
  28:  (11100) = (11100)
  30:  (11110) = (11110)
  31:  (11111) = (1)(1)(1)(1)(1)
  32: (100000) = (100000)
  36: (100100) = (100)(100)
  38: (100110) = (100)(110)
  40: (101000) = (101000)
  42: (101010) = (10)(10)(10)
		

Crossrefs

Numbers whose binary expansion has uniform Lyndon factorization are A023758.

Numbers whose reversed binary expansion is Lyndon are A328596.

Numbers whose binary expansion is co-Lyndon are A275692.

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

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

Programs

Mathematica

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Select[Range[100],SameQ@@Length/@colynfac[IntegerDigits[#,2]]&]