cp's OEIS Frontend

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

%I A329327 #21 Mar 31 2021 01:47:40
%S A329327 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 A329327 131,133,135,137,139,141,143,147,149,151,155,157,159,171,175,183,191,
%U A329327 257,259,261,263,265,267,269,271,275,277,279,281,283,285,287,293
%N A329327 Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).
%C A329327 First differs from A329357 in having 77 and lacking 83.
%C A329327 Also numbers whose decapitated binary expansion is a Lyndon word (see also A329401).
%F A329327 a(n) = A339608(n) + 1. - _Harald Korneliussen_, Mar 12 2020
%e A329327 The binary expansion of each term together with its Lyndon factorization begins:
%e A329327    2:      (10) = (1)(0)
%e A329327    3:      (11) = (1)(1)
%e A329327    5:     (101) = (1)(01)
%e A329327    9:    (1001) = (1)(001)
%e A329327   11:    (1011) = (1)(011)
%e A329327   17:   (10001) = (1)(0001)
%e A329327   19:   (10011) = (1)(0011)
%e A329327   23:   (10111) = (1)(0111)
%e A329327   33:  (100001) = (1)(00001)
%e A329327   35:  (100011) = (1)(00011)
%e A329327   37:  (100101) = (1)(00101)
%e A329327   39:  (100111) = (1)(00111)
%e A329327   43:  (101011) = (1)(01011)
%e A329327   47:  (101111) = (1)(01111)
%e A329327   65: (1000001) = (1)(000001)
%e A329327   67: (1000011) = (1)(000011)
%e A329327   69: (1000101) = (1)(000101)
%e A329327   71: (1000111) = (1)(000111)
%e A329327   75: (1001011) = (1)(001011)
%e A329327   77: (1001101) = (1)(001101)
%t A329327 lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
%t A329327 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 A329327 Select[Range[100],Length[lynfac[IntegerDigits[#,2]]]==2&]
%Y A329327 Positions of 2's in A211100.
%Y A329327 Positions of rows of length 2 in A329314.
%Y A329327 The "co-" and reversed version is A329357.
%Y A329327 Binary Lyndon words are counted by A001037 and ranked by A102659.
%Y A329327 Length of the co-Lyndon factorization of the binary expansion is A329312.
%Y A329327 Cf. A059966, A060223, A211097, A275692, A328594, A328595, A328596, A329131, A329313, A329325, A329326, A339608.
%K A329327 nonn
%O A329327 1,1
%A A329327 _Gus Wiseman_, Nov 12 2019