A328594 - OEIS

%I #7 May 01 2020 08:01:41

%S 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,

%T 29,30,32,33,34,35,37,38,39,40,41,43,44,46,47,48,49,50,51,52,53,55,56,

%U 57,58,59,60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,77

%N Numbers whose binary expansion is aperiodic.

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

%C 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

%e The sequence of terms together with their binary expansions and binary indices begins:

%e    0:     0 ~ {}

%e    1:     1 ~ {1}

%e    2:    10 ~ {2}

%e    4:   100 ~ {3}

%e    5:   101 ~ {1,3}

%e    6:   110 ~ {2,3}

%e    8:  1000 ~ {4}

%e    9:  1001 ~ {1,4}

%e   11:  1011 ~ {1,2,4}

%e   12:  1100 ~ {3,4}

%e   13:  1101 ~ {1,3,4}

%e   14:  1110 ~ {2,3,4}

%e   16: 10000 ~ {5}

%e   17: 10001 ~ {1,5}

%e   18: 10010 ~ {2,5}

%e   19: 10011 ~ {1,2,5}

%e   20: 10100 ~ {3,5}

%e   21: 10101 ~ {1,3,5}

%e   22: 10110 ~ {2,3,5}

%e   23: 10111 ~ {1,2,3,5}

%e   24: 11000 ~ {4,5}

%t aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];

%t Select[Range[0,100],aperQ[IntegerDigits[#,2]]&]

%Y The complement is A121016.

%Y The version for prime indices is A085971.

%Y Numbers without proper integer roots are A007916.

%Y Necklaces are A328595.

%Y Lyndon words are A328596.

%Y Aperiodic compositions are A000740.

%Y Aperiodic binary sequences are A027375.

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Oct 22 2019