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A326774 For any number m, let m* be the bi-infinite string obtained by repetition of the binary representation of m; this sequence lists the numbers n such that for any k < n, n* does not equal k* up to a shift.

%I A326774 #13 Apr 19 2020 12:44:18
%S A326774 0,1,2,4,5,8,9,11,16,17,18,19,21,23,32,33,34,35,37,38,39,43,47,64,65,
%T A326774 66,67,68,69,70,71,73,74,75,77,78,79,85,87,91,95,128,129,130,131,132,
%U A326774 133,134,135,137,138,139,140,141,142,143,146,147,149,150,151,154
%N A326774 For any number m, let m* be the bi-infinite string obtained by repetition of the binary representation of m; this sequence lists the numbers n such that for any k < n, n* does not equal k* up to a shift.
%C A326774 This sequence contains every power of 2.
%C A326774 No term belongs to A121016.
%C A326774 Every terms belongs to A004761.
%C A326774 For any k > 0, there are A001037(k) terms with binary length k.
%C A326774 From _Gus Wiseman_, Apr 19 2020: (Start)
%C A326774 Also numbers k such that the k-th composition in standard order (row k of A066099) is a co-Lyndon word (regular Lyndon words being A275692). For example, the sequence of all co-Lyndon words begins:
%C A326774     0: ()            37: (3,2,1)         79: (3,1,1,1,1)
%C A326774     1: (1)           38: (3,1,2)         85: (2,2,2,1)
%C A326774     2: (2)           39: (3,1,1,1)       87: (2,2,1,1,1)
%C A326774     4: (3)           43: (2,2,1,1)       91: (2,1,2,1,1)
%C A326774     5: (2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
%C A326774     8: (4)           64: (7)            128: (8)
%C A326774     9: (3,1)         65: (6,1)          129: (7,1)
%C A326774    11: (2,1,1)       66: (5,2)          130: (6,2)
%C A326774    16: (5)           67: (5,1,1)        131: (6,1,1)
%C A326774    17: (4,1)         68: (4,3)          132: (5,3)
%C A326774    18: (3,2)         69: (4,2,1)        133: (5,2,1)
%C A326774    19: (3,1,1)       70: (4,1,2)        134: (5,1,2)
%C A326774    21: (2,2,1)       71: (4,1,1,1)      135: (5,1,1,1)
%C A326774    23: (2,1,1,1)     73: (3,3,1)        137: (4,3,1)
%C A326774    32: (6)           74: (3,2,2)        138: (4,2,2)
%C A326774    33: (5,1)         75: (3,2,1,1)      139: (4,2,1,1)
%C A326774    34: (4,2)         77: (3,1,2,1)      140: (4,1,3)
%C A326774    35: (4,1,1)       78: (3,1,1,2)      141: (4,1,2,1)
%C A326774 (End)
%H A326774 Rémy Sigrist, <a href="/A326774/a326774.png">Logarithmic scatterplot of the first differences of the terms < 2^24</a>
%H A326774 Rémy Sigrist, <a href="/A326774/a326774.gp.txt">PARI program for A326774</a>
%e A326774 3* = ...11... equals 1* = ...1..., so 3 is not a term.
%e A326774 6* = ...110... equals up to a shift 5* = ...101..., so 6 is not a term.
%e A326774 11* = ...1011... only equals up to a shift 13* = ...1101... and 14* = ...1110..., so 11 is a term.
%t A326774 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A326774 colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A326774 Select[Range[0,100],colynQ[stc[#]]&] (* _Gus Wiseman_, Apr 19 2020 *)
%o A326774 (PARI) See Links section.
%Y A326774 Cf. A001037, A004761, A065609, A121016.
%Y A326774 Necklace compositions are counted by A008965.
%Y A326774 Lyndon compositions are counted by A059966.
%Y A326774 Length of Lyndon factorization of binary expansion is A211100.
%Y A326774 Numbers whose reversed binary expansion is a necklace are A328595.
%Y A326774 Length of co-Lyndon factorization of binary expansion is A329312.
%Y A326774 Length of Lyndon factorization of reversed binary expansion is A329313.
%Y A326774 Length of co-Lyndon factorization of reversed binary expansion is A329326.
%Y A326774 All of the following pertain to compositions in standard order (A066099):
%Y A326774 - Length is A000120.
%Y A326774 - Necklaces are A065609.
%Y A326774 - Sum is A070939.
%Y A326774 - Runs are counted by A124767.
%Y A326774 - Rotational symmetries are counted by A138904.
%Y A326774 - Strict compositions are A233564.
%Y A326774 - Constant compositions are A272919.
%Y A326774 - Lyndon compositions are A275692.
%Y A326774 - Co-Lyndon compositions are A326774 (this sequence).
%Y A326774 - Aperiodic compositions are A328594.
%Y A326774 - Reversed co-necklaces are A328595.
%Y A326774 - Rotational period is A333632.
%Y A326774 - Co-necklaces are A333764.
%Y A326774 - Co-Lyndon factorizations are counted by A333765.
%Y A326774 - Lyndon factorizations are counted by A333940.
%Y A326774 - Reversed necklaces are A333943.
%Y A326774 - Length of co-Lyndon factorization is A334029.
%Y A326774 Cf. A000740, A034691, A060223, A269134, A292884, A328596.
%K A326774 nonn,base
%O A326774 0,3
%A A326774 _Rémy Sigrist_, Jul 27 2019