cp's OEIS Frontend

A329399 Numbers whose reversed binary expansion has uniform Lyndon factorization.

%I A329399 #4 Nov 14 2019 12:59:05
%S A329399 1,2,3,4,6,7,8,10,12,14,15,16,20,24,26,28,30,31,32,36,38,40,42,44,48,
%T A329399 52,54,56,58,60,62,63,64,72,80,84,88,92,96,100,104,106,108,112,116,
%U A329399 118,120,122,124,126,127,128,136,140,142,144,152,160,164,168,170
%N A329399 Numbers whose reversed binary expansion has uniform Lyndon factorization.
%C A329399 We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
%C A329399 A sequence of words is uniform if they all have the same length.
%e A329399 The sequence of terms together with their reversed binary expansions and Lyndon factorizations begins:
%e A329399    1:      (1) = (1)
%e A329399    2:     (01) = (01)
%e A329399    3:     (11) = (1)(1)
%e A329399    4:    (001) = (001)
%e A329399    6:    (011) = (011)
%e A329399    7:    (111) = (1)(1)(1)
%e A329399    8:   (0001) = (0001)
%e A329399   10:   (0101) = (01)(01)
%e A329399   12:   (0011) = (0011)
%e A329399   14:   (0111) = (0111)
%e A329399   15:   (1111) = (1)(1)(1)(1)
%e A329399   16:  (00001) = (00001)
%e A329399   20:  (00101) = (00101)
%e A329399   24:  (00011) = (00011)
%e A329399   26:  (01011) = (01011)
%e A329399   28:  (00111) = (00111)
%e A329399   30:  (01111) = (01111)
%e A329399   31:  (11111) = (1)(1)(1)(1)(1)
%e A329399   32: (000001) = (000001)
%e A329399   36: (001001) = (001)(001)
%e A329399   38: (011001) = (011)(001)
%e A329399   40: (000101) = (000101)
%e A329399   42: (010101) = (01)(01)(01)
%e A329399   44: (001101) = (001101)
%e A329399   48: (000011) = (000011)
%t A329399 lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
%t A329399 lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
%t A329399 Select[Range[100],SameQ@@Length/@lynfac[Reverse[IntegerDigits[#,2]]]&]
%Y A329399 Numbers whose binary expansion has uniform Lyndon factorization and uniform co-Lyndon factorization are A023758.
%Y A329399 Numbers whose reversed binary expansion is Lyndon are A328596.
%Y A329399 Numbers whose binary expansion is co-Lyndon are A275692.
%Y A329399 Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
%Y A329399 Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329396.
%K A329399 nonn
%O A329399 1,2
%A A329399 _Gus Wiseman_, Nov 13 2019