A329327 Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).

2, 3, 5, 9, 11, 17, 19, 23, 33, 35, 37, 39, 43, 47, 65, 67, 69, 71, 75, 77, 79, 87, 95, 129, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 155, 157, 159, 171, 175, 183, 191, 257, 259, 261, 263, 265, 267, 269, 271, 275, 277, 279, 281, 283, 285, 287, 293

Offset: 1

Author

Gus Wiseman, Nov 12 2019

Keywords

• nonn

Comments

First differs from A329357 in having 77 and lacking 83.

Also numbers whose decapitated binary expansion is a Lyndon word (see also A329401).

Examples

			The binary expansion of each term together with its Lyndon factorization begins:
   2:      (10) = (1)(0)
   3:      (11) = (1)(1)
   5:     (101) = (1)(01)
   9:    (1001) = (1)(001)
  11:    (1011) = (1)(011)
  17:   (10001) = (1)(0001)
  19:   (10011) = (1)(0011)
  23:   (10111) = (1)(0111)
  33:  (100001) = (1)(00001)
  35:  (100011) = (1)(00011)
  37:  (100101) = (1)(00101)
  39:  (100111) = (1)(00111)
  43:  (101011) = (1)(01011)
  47:  (101111) = (1)(01111)
  65: (1000001) = (1)(000001)
  67: (1000011) = (1)(000011)
  69: (1000101) = (1)(000101)
  71: (1000111) = (1)(000111)
  75: (1001011) = (1)(001011)
  77: (1001101) = (1)(001101)
		

Crossrefs

Positions of 2's in A211100.

Positions of rows of length 2 in A329314.

The "co-" and reversed version is A329357.

Binary Lyndon words are counted by A001037 and ranked by A102659.

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

Cf. A059966, A060223, A211097, A275692, A328594, A328595, A328596, A329131, A329313, A329325, A329326, A339608.

Programs

Mathematica

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

Formula

a(n) = A339608(n) + 1. - Harald Korneliussen, Mar 12 2020