A333943 Numbers k such that the k-th composition in standard order is a reversed 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, 39, 41, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 141, 143

Offset: 1

Author

Gus Wiseman, Apr 14 2020

Keywords

• nonn

Comments

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations. Reversed necklaces are different from co-necklaces (A333764).

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 reversed necklaces begins:
    1: (1)             32: (6)               69: (4,2,1)
    2: (2)             33: (5,1)             71: (4,1,1,1)
    3: (1,1)           34: (4,2)             73: (3,3,1)
    4: (3)             35: (4,1,1)           74: (3,2,2)
    5: (2,1)           36: (3,3)             75: (3,2,1,1)
    7: (1,1,1)         37: (3,2,1)           77: (3,1,2,1)
    8: (4)             39: (3,1,1,1)         79: (3,1,1,1,1)
    9: (3,1)           41: (2,3,1)           81: (2,4,1)
   10: (2,2)           42: (2,2,2)           83: (2,3,1,1)
   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-reversed version is A065609.

The dual 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.

- Co-necklaces are A333764.

- Length of co-Lyndon factorization is A334029.

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;
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#1]}]&,Length[q]-1,1,And];
Select[Range[100],neckQ[Reverse[stc[#]]]&]