A329357 Numbers whose reversed binary expansion has co-Lyndon factorization of length 2.

2, 3, 5, 9, 11, 17, 19, 23, 33, 35, 37, 39, 43, 47, 65, 67, 69, 71, 75, 79, 83, 87, 95, 129, 131, 133, 135, 137, 139, 143, 147, 149, 151, 155, 159, 163, 167, 171, 175, 183, 191, 257, 259, 261, 263, 265, 267, 271, 275, 277, 279, 283, 287, 291, 293, 295, 299

Offset

1,1

Comments

First differs from A329327 in lacking 77 and having 83.

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

Links

Table of n, a(n) for n=1..57.

Example

The reversed binary expansion of each term together with their co-Lyndon factorizations:
   2:      (01) = (0)(1)
   3:      (11) = (1)(1)
   5:     (101) = (10)(1)
   9:    (1001) = (100)(1)
  11:    (1101) = (110)(1)
  17:   (10001) = (1000)(1)
  19:   (11001) = (1100)(1)
  23:   (11101) = (1110)(1)
  33:  (100001) = (10000)(1)
  35:  (110001) = (11000)(1)
  37:  (101001) = (10100)(1)
  39:  (111001) = (11100)(1)
  43:  (110101) = (11010)(1)
  47:  (111101) = (11110)(1)
  65: (1000001) = (100000)(1)
  67: (1100001) = (110000)(1)
  69: (1010001) = (101000)(1)
  71: (1110001) = (111000)(1)
  75: (1101001) = (110100)(1)
  79: (1111001) = (111100)(1)

Mathematica

colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Select[Range[100], Length[colynfac[Reverse[IntegerDigits[#, 2]]]]==2&]

Crossrefs

Positions of 2's in A329326.

Binary co-Lyndon words are counted by A001037 and ranked by A329318.

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

Cf. A059966, A060223, A102659, A211097, A211100, A275692, A328594, A328595, A328596, A329131, A329313, A329314, A329325, A329359.

Sequence in context: A157604 A238657 A329327 * A101737 A176550 A364059

Adjacent sequences: A329354 A329355 A329356 * A329358 A329359 A329360

Keyword

nonn

Author

Gus Wiseman, Nov 12 2019

Status

approved