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