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Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).
6

%I #21 Mar 31 2021 01:47:40

%S 2,3,5,9,11,17,19,23,33,35,37,39,43,47,65,67,69,71,75,77,79,87,95,129,

%T 131,133,135,137,139,141,143,147,149,151,155,157,159,171,175,183,191,

%U 257,259,261,263,265,267,269,271,275,277,279,281,283,285,287,293

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

%C First differs from A329357 in having 77 and lacking 83.

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

%F a(n) = A339608(n) + 1. - _Harald Korneliussen_, Mar 12 2020

%e The binary expansion of each term together with its Lyndon factorization begins:

%e 2: (10) = (1)(0)

%e 3: (11) = (1)(1)

%e 5: (101) = (1)(01)

%e 9: (1001) = (1)(001)

%e 11: (1011) = (1)(011)

%e 17: (10001) = (1)(0001)

%e 19: (10011) = (1)(0011)

%e 23: (10111) = (1)(0111)

%e 33: (100001) = (1)(00001)

%e 35: (100011) = (1)(00011)

%e 37: (100101) = (1)(00101)

%e 39: (100111) = (1)(00111)

%e 43: (101011) = (1)(01011)

%e 47: (101111) = (1)(01111)

%e 65: (1000001) = (1)(000001)

%e 67: (1000011) = (1)(000011)

%e 69: (1000101) = (1)(000101)

%e 71: (1000111) = (1)(000111)

%e 75: (1001011) = (1)(001011)

%e 77: (1001101) = (1)(001101)

%t lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];

%t 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]]&]]]];

%t Select[Range[100],Length[lynfac[IntegerDigits[#,2]]]==2&]

%Y Positions of 2's in A211100.

%Y Positions of rows of length 2 in A329314.

%Y The "co-" and reversed version is A329357.

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

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 12 2019