A333764 Numbers k such that the k-th composition in standard order is a co-necklace.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140

Offset: 1

Author

Gus Wiseman, Apr 12 2020

Keywords

• nonn

Comments

A co-necklace is a finite sequence that is lexicographically greater than or equal to any cyclic rotation.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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

Examples

			The sequence together with the corresponding co-necklaces begins:
    1: (1)             32: (6)               69: (4,2,1)
    2: (2)             33: (5,1)             70: (4,1,2)
    3: (1,1)           34: (4,2)             71: (4,1,1,1)
    4: (3)             35: (4,1,1)           73: (3,3,1)
    5: (2,1)           36: (3,3)             74: (3,2,2)
    7: (1,1,1)         37: (3,2,1)           75: (3,2,1,1)
    8: (4)             38: (3,1,2)           77: (3,1,2,1)
    9: (3,1)           39: (3,1,1,1)         78: (3,1,1,2)
   10: (2,2)           42: (2,2,2)           79: (3,1,1,1,1)
   11: (2,1,1)         43: (2,2,1,1)         85: (2,2,2,1)
   15: (1,1,1,1)       45: (2,1,2,1)         87: (2,2,1,1,1)
   16: (5)             47: (2,1,1,1,1)       91: (2,1,2,1,1)
   17: (4,1)           63: (1,1,1,1,1,1)     95: (2,1,1,1,1,1)
   18: (3,2)           64: (7)              127: (1,1,1,1,1,1,1)
   19: (3,1,1)         65: (6,1)            128: (8)
   21: (2,2,1)         66: (5,2)            129: (7,1)
   23: (2,1,1,1)       67: (5,1,1)          130: (6,2)
   31: (1,1,1,1,1)     68: (4,3)            131: (6,1,1)
		

Crossrefs

The non-"co" version is A065609.

The reversed version is A328595.

Binary necklaces are A000031.

Necklace compositions are A008965.

Necklaces covering an initial interval are A019536.

Numbers whose prime signature is a necklace are A329138.

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

Length of Lyndon factorization of reversed binary expansion is A329313.

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

- Length is A000120.

- 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.

- Aperiodic compositions are A328594.

- Length of Lyndon factorization is A329312.

- Rotational period is A333632.

- Reversed necklaces are A333943.

Cf. A000740, A001037, A027375, A059966, A211100, A302291, A328596, A329142, A333765, A333939, A333941.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
coneckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[100],coneckQ[stc[#]]&]