cp's OEIS Frontend

A328594 Numbers whose binary expansion is aperiodic.

%I A328594 #7 May 01 2020 08:01:41
%S A328594 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 A328594 29,30,32,33,34,35,37,38,39,40,41,43,44,46,47,48,49,50,51,52,53,55,56,
%U A328594 57,58,59,60,61,62,64,65,66,67,68,69,70,71,72,73,74,75,76,77
%N A328594 Numbers whose binary expansion is aperiodic.
%C A328594 A finite sequence is aperiodic if all of its cyclic rotations are distinct. See A000740 or A027375 for details.
%C A328594 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 A328594 The sequence of terms together with their binary expansions and binary indices begins:
%e A328594    0:     0 ~ {}
%e A328594    1:     1 ~ {1}
%e A328594    2:    10 ~ {2}
%e A328594    4:   100 ~ {3}
%e A328594    5:   101 ~ {1,3}
%e A328594    6:   110 ~ {2,3}
%e A328594    8:  1000 ~ {4}
%e A328594    9:  1001 ~ {1,4}
%e A328594   11:  1011 ~ {1,2,4}
%e A328594   12:  1100 ~ {3,4}
%e A328594   13:  1101 ~ {1,3,4}
%e A328594   14:  1110 ~ {2,3,4}
%e A328594   16: 10000 ~ {5}
%e A328594   17: 10001 ~ {1,5}
%e A328594   18: 10010 ~ {2,5}
%e A328594   19: 10011 ~ {1,2,5}
%e A328594   20: 10100 ~ {3,5}
%e A328594   21: 10101 ~ {1,3,5}
%e A328594   22: 10110 ~ {2,3,5}
%e A328594   23: 10111 ~ {1,2,3,5}
%e A328594   24: 11000 ~ {4,5}
%t A328594 aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
%t A328594 Select[Range[0,100],aperQ[IntegerDigits[#,2]]&]
%Y A328594 The complement is A121016.
%Y A328594 The version for prime indices is A085971.
%Y A328594 Numbers without proper integer roots are A007916.
%Y A328594 Necklaces are A328595.
%Y A328594 Lyndon words are A328596.
%Y A328594 Aperiodic compositions are A000740.
%Y A328594 Aperiodic binary sequences are A027375.
%Y A328594 Cf. A000120, A000939, A014081, A065609, A069010, A275692, A323867, A334030.
%K A328594 nonn
%O A328594 1,3
%A A328594 _Gus Wiseman_, Oct 22 2019