A328596 Numbers whose reversed binary expansion is a Lyndon word (aperiodic necklace).

1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 26, 28, 30, 32, 40, 44, 48, 52, 56, 58, 60, 62, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 128, 144, 152, 160, 164, 168, 172, 176, 180, 184, 188, 192, 200, 208, 212, 216, 218, 220, 224

Offset: 1

Author

Gus Wiseman, Oct 22 2019

Keywords

• nonn
• base

Comments

First differs from A091065 in lacking 50.

A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.

Examples

			The sequence of terms together with their binary expansions and binary indices begins:
   1:      1 ~ {1}
   2:     10 ~ {2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   8:   1000 ~ {4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  16:  10000 ~ {5}
  20:  10100 ~ {3,5}
  24:  11000 ~ {4,5}
  26:  11010 ~ {2,4,5}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  32: 100000 ~ {6}
  40: 101000 ~ {4,6}
  44: 101100 ~ {3,4,6}
  48: 110000 ~ {5,6}
  52: 110100 ~ {3,5,6}
  56: 111000 ~ {4,5,6}
  58: 111010 ~ {2,4,5,6}
		

Crossrefs

A similar concept is A275692.

Aperiodic words are A328594.

Necklaces are A328595.

Binary Lyndon words are A001037.

Lyndon compositions are A059966.

Cf. A000031, A000120, A000740, A008965, A027375, A121016.

Programs

Mathematica

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],aperQ[Reverse[IntegerDigits[#,2]]]&&neckQ[Reverse[IntegerDigits[#,2]]]&]

Formula

Intersection of A328594 and A328595.