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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.
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%I #13 Apr 19 2020 12:44:18

%S 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 66,67,68,69,70,71,73,74,75,77,78,79,85,87,91,95,128,129,130,131,132,

%U 133,134,135,137,138,139,140,141,142,143,146,147,149,150,151,154

%N 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 This sequence contains every power of 2.

%C No term belongs to A121016.

%C Every terms belongs to A004761.

%C For any k > 0, there are A001037(k) terms with binary length k.

%C From _Gus Wiseman_, Apr 19 2020: (Start)

%C 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 0: () 37: (3,2,1) 79: (3,1,1,1,1)

%C 1: (1) 38: (3,1,2) 85: (2,2,2,1)

%C 2: (2) 39: (3,1,1,1) 87: (2,2,1,1,1)

%C 4: (3) 43: (2,2,1,1) 91: (2,1,2,1,1)

%C 5: (2,1) 47: (2,1,1,1,1) 95: (2,1,1,1,1,1)

%C 8: (4) 64: (7) 128: (8)

%C 9: (3,1) 65: (6,1) 129: (7,1)

%C 11: (2,1,1) 66: (5,2) 130: (6,2)

%C 16: (5) 67: (5,1,1) 131: (6,1,1)

%C 17: (4,1) 68: (4,3) 132: (5,3)

%C 18: (3,2) 69: (4,2,1) 133: (5,2,1)

%C 19: (3,1,1) 70: (4,1,2) 134: (5,1,2)

%C 21: (2,2,1) 71: (4,1,1,1) 135: (5,1,1,1)

%C 23: (2,1,1,1) 73: (3,3,1) 137: (4,3,1)

%C 32: (6) 74: (3,2,2) 138: (4,2,2)

%C 33: (5,1) 75: (3,2,1,1) 139: (4,2,1,1)

%C 34: (4,2) 77: (3,1,2,1) 140: (4,1,3)

%C 35: (4,1,1) 78: (3,1,1,2) 141: (4,1,2,1)

%C (End)

%H Rémy Sigrist, <a href="/A326774/a326774.png">Logarithmic scatterplot of the first differences of the terms < 2^24</a>

%H Rémy Sigrist, <a href="/A326774/a326774.gp.txt">PARI program for A326774</a>

%e 3* = ...11... equals 1* = ...1..., so 3 is not a term.

%e 6* = ...110... equals up to a shift 5* = ...101..., so 6 is not a term.

%e 11* = ...1011... only equals up to a shift 13* = ...1101... and 14* = ...1110..., so 11 is a term.

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];

%t Select[Range[0,100],colynQ[stc[#]]&] (* _Gus Wiseman_, Apr 19 2020 *)

%o (PARI) See Links section.

%Y Cf. A001037, A004761, A065609, A121016.

%Y Necklace compositions are counted by A008965.

%Y Lyndon compositions are counted by A059966.

%Y Length of Lyndon factorization of binary expansion is A211100.

%Y Numbers whose reversed binary expansion is a necklace are A328595.

%Y Length of co-Lyndon factorization of binary expansion is A329312.

%Y Length of Lyndon factorization of reversed binary expansion is A329313.

%Y Length of co-Lyndon factorization of reversed binary expansion is A329326.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Necklaces are A065609.

%Y - Sum is A070939.

%Y - Runs are counted by A124767.

%Y - Rotational symmetries are counted by A138904.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Lyndon compositions are A275692.

%Y - Co-Lyndon compositions are A326774 (this sequence).

%Y - Aperiodic compositions are A328594.

%Y - Reversed co-necklaces are A328595.

%Y - Rotational period is A333632.

%Y - Co-necklaces are A333764.

%Y - Co-Lyndon factorizations are counted by A333765.

%Y - Lyndon factorizations are counted by A333940.

%Y - Reversed necklaces are A333943.

%Y - Length of co-Lyndon factorization is A334029.

%Y Cf. A000740, A034691, A060223, A269134, A292884, A328596.

%K nonn,base

%O 0,3

%A _Rémy Sigrist_, Jul 27 2019