cp's OEIS Frontend

A329357 Numbers whose reversed binary expansion has co-Lyndon factorization of length 2.

%I A329357 #9 Nov 15 2019 09:35:46
%S A329357 2,3,5,9,11,17,19,23,33,35,37,39,43,47,65,67,69,71,75,79,83,87,95,129,
%T A329357 131,133,135,137,139,143,147,149,151,155,159,163,167,171,175,183,191,
%U A329357 257,259,261,263,265,267,271,275,277,279,283,287,291,293,295,299
%N A329357 Numbers whose reversed binary expansion has co-Lyndon factorization of length 2.
%C A329357 First differs from A329327 in lacking 77 and having 83.
%C A329357 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).
%e A329357 The reversed binary expansion of each term together with their co-Lyndon factorizations:
%e A329357    2:      (01) = (0)(1)
%e A329357    3:      (11) = (1)(1)
%e A329357    5:     (101) = (10)(1)
%e A329357    9:    (1001) = (100)(1)
%e A329357   11:    (1101) = (110)(1)
%e A329357   17:   (10001) = (1000)(1)
%e A329357   19:   (11001) = (1100)(1)
%e A329357   23:   (11101) = (1110)(1)
%e A329357   33:  (100001) = (10000)(1)
%e A329357   35:  (110001) = (11000)(1)
%e A329357   37:  (101001) = (10100)(1)
%e A329357   39:  (111001) = (11100)(1)
%e A329357   43:  (110101) = (11010)(1)
%e A329357   47:  (111101) = (11110)(1)
%e A329357   65: (1000001) = (100000)(1)
%e A329357   67: (1100001) = (110000)(1)
%e A329357   69: (1010001) = (101000)(1)
%e A329357   71: (1110001) = (111000)(1)
%e A329357   75: (1101001) = (110100)(1)
%e A329357   79: (1111001) = (111100)(1)
%t A329357 colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A329357 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 A329357 Select[Range[100],Length[colynfac[Reverse[IntegerDigits[#,2]]]]==2&]
%Y A329357 Positions of 2's in A329326.
%Y A329357 Binary co-Lyndon words are counted by A001037 and ranked by A329318.
%Y A329357 Length of the co-Lyndon factorization of the binary expansion is A329312.
%Y A329357 Cf. A059966, A060223, A102659, A211097, A211100, A275692, A328594, A328595, A328596, A329131, A329313, A329314, A329325, A329359.
%K A329357 nonn
%O A329357 1,1
%A A329357 _Gus Wiseman_, Nov 12 2019