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.

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, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 146, 147, 149, 150, 151, 154

Offset: 0

Author

Rémy Sigrist, Jul 27 2019

Keywords

• nonn
• base

Comments

This sequence contains every power of 2.

No term belongs to A121016.

Every terms belongs to A004761.

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

From Gus Wiseman, Apr 19 2020: (Start)

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(End)

Examples

			3* = ...11... equals 1* = ...1..., so 3 is not a term.
6* = ...110... equals up to a shift 5* = ...101..., so 6 is not a term.
11* = ...1011... only equals up to a shift 13* = ...1101... and 14* = ...1110..., so 11 is a term.
		

Links

• Rémy Sigrist, Logarithmic scatterplot of the first differences of the terms < 2^24
• Rémy Sigrist, PARI program for A326774

Crossrefs

Cf. A001037, A004761, A065609, A121016.

Necklace compositions are counted by A008965.

Lyndon compositions are counted by A059966.

Length of Lyndon factorization of binary expansion is A211100.

Numbers whose reversed binary expansion is a necklace are A328595.

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

Length of Lyndon factorization of reversed binary expansion is A329313.

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

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

- Length is A000120.

- Necklaces are A065609.

- Sum is A070939.

- Runs are counted by A124767.

- Rotational symmetries are counted by A138904.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Lyndon compositions are A275692.

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

- Aperiodic compositions are A328594.

- Reversed co-necklaces are A328595.

- Rotational period is A333632.

- Co-necklaces are A333764.

- Co-Lyndon factorizations are counted by A333765.

- Lyndon factorizations are counted by A333940.

- Reversed necklaces are A333943.

- Length of co-Lyndon factorization is A334029.

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

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],colynQ[stc[#]]&] (* Gus Wiseman, Apr 19 2020 *)

PARI

See Links section.