A328594 Numbers whose binary expansion is aperiodic.

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset

1,3

Comments

A finite sequence is aperiodic if all of its cyclic rotations are distinct. See A000740 or A027375 for details.

Also numbers k such that the k-th composition in standard order is aperiodic. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 28 2020

Links

Table of n, a(n) for n=1..68.

Example

The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}

Mathematica

aperQ[q_]:=Array[RotateRight[q, #]&, Length[q], 1, UnsameQ];
Select[Range[0, 100], aperQ[IntegerDigits[#, 2]]&]

Crossrefs

The complement is A121016.

The version for prime indices is A085971.

Numbers without proper integer roots are A007916.

Necklaces are A328595.

Lyndon words are A328596.

Aperiodic compositions are A000740.

Aperiodic binary sequences are A027375.

Cf. A000120, A000939, A014081, A065609, A069010, A275692, A323867, A334030.

Sequence in context: A023717 A324639 A171599 * A346129 A366160 A288174

Adjacent sequences: A328591 A328592 A328593 * A328595 A328596 A328597

Keyword

nonn

Author

Gus Wiseman, Oct 22 2019

Status

approved