cp's OEIS Frontend

A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

%I A329396 #9 Jul 02 2021 01:55:22
%S A329396 1,2,3,4,6,7,8,10,12,14,15,16,20,24,26,28,30,31,32,36,38,40,42,48,50,
%T A329396 52,54,56,58,60,62,63,64,72,80,84,96,98,100,104,106,108,112,114,116,
%U A329396 118,120,122,124,126,127,128,136,140,142,144,160,164,168,170,192
%N A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.
%C A329396 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).
%C A329396 A sequence of words is uniform if they all have the same length.
%e A329396 The sequence of terms together with their co-Lyndon factorizations begins:
%e A329396    1:      (1) = (1)
%e A329396    2:     (10) = (10)
%e A329396    3:     (11) = (1)(1)
%e A329396    4:    (100) = (100)
%e A329396    6:    (110) = (110)
%e A329396    7:    (111) = (1)(1)(1)
%e A329396    8:   (1000) = (1000)
%e A329396   10:   (1010) = (10)(10)
%e A329396   12:   (1100) = (1100)
%e A329396   14:   (1110) = (1110)
%e A329396   15:   (1111) = (1)(1)(1)(1)
%e A329396   16:  (10000) = (10000)
%e A329396   20:  (10100) = (10100)
%e A329396   24:  (11000) = (11000)
%e A329396   26:  (11010) = (11010)
%e A329396   28:  (11100) = (11100)
%e A329396   30:  (11110) = (11110)
%e A329396   31:  (11111) = (1)(1)(1)(1)(1)
%e A329396   32: (100000) = (100000)
%e A329396   36: (100100) = (100)(100)
%e A329396   38: (100110) = (100)(110)
%e A329396   40: (101000) = (101000)
%e A329396   42: (101010) = (10)(10)(10)
%t A329396 colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A329396 colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
%t A329396 Select[Range[100],SameQ@@Length/@colynfac[IntegerDigits[#,2]]&]
%Y A329396 Numbers whose binary expansion has uniform Lyndon factorization are A023758.
%Y A329396 Numbers whose reversed binary expansion is Lyndon are A328596.
%Y A329396 Numbers whose binary expansion is co-Lyndon are A275692.
%Y A329396 Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
%Y A329396 Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329398.
%K A329396 nonn
%O A329396 1,2
%A A329396 _Gus Wiseman_, Nov 13 2019