A328668 Numbers whose binary expansion, without the most significant digit, is a necklace.

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 21, 23, 31, 32, 33, 35, 37, 39, 43, 47, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 85, 87, 91, 95, 127, 128, 129, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 155, 157, 159, 171, 175, 183, 191, 255, 256, 257

Offset

0,3

Comments

Offset is 0 to be consistent with A257250.

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

Links

Table of n, a(n) for n=0..60.

Example

The sequence of terms together with their binary expansions and binary indices begins:
   0:       0 ~ {}
   1:       1 ~ {1}
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   4:     100 ~ {3}
   5:     101 ~ {1,3}
   7:     111 ~ {1,2,3}
   8:    1000 ~ {4}
   9:    1001 ~ {1,4}
  11:    1011 ~ {1,2,4}
  15:    1111 ~ {1,2,3,4}
  16:   10000 ~ {5}
  17:   10001 ~ {1,5}
  19:   10011 ~ {1,2,5}
  21:   10101 ~ {1,3,5}
  23:   10111 ~ {1,2,3,5}
  31:   11111 ~ {1,2,3,4,5}
  32:  100000 ~ {6}
  33:  100001 ~ {1,6}
  35:  100011 ~ {1,2,6}

Mathematica

neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Select[Range[0, 100], #<=1||neckQ[Rest[IntegerDigits[#, 2]]]&]

Crossrefs

The dual version is A257250.

The version involving all digits, taken in reverse, is A328595.

The reversed version is A328607.

Binary necklaces are A000031.

Necklace compositions are A008965.

Cf. A000120, A001037, A003714, A014081, A065609, A121016, A275692, A328594, A328596.

Sequence in context: A282136 A153730 A140691 * A316468 A099627 A324841

Adjacent sequences: A328665 A328666 A328667 * A328669 A328670 A328671

Keyword

nonn

Author

Gus Wiseman, Oct 26 2019

Status

approved