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A275692 Numbers k such that every rotation of the binary digits of k is less than k.

%I A275692 #27 Nov 26 2022 16:14:47
%S A275692 0,1,2,4,6,8,12,14,16,20,24,26,28,30,32,40,48,50,52,56,58,60,62,64,72,
%T A275692 80,84,96,98,100,104,106,108,112,114,116,118,120,122,124,126,128,144,
%U A275692 160,164,168,192,194,196,200,202,208,210,212,216,218,224,226,228
%N A275692 Numbers k such that every rotation of the binary digits of k is less than k.
%C A275692 0, and terms of A065609 that are not in A121016.
%C A275692 Number of terms with d binary digits is A001037(d).
%C A275692 Take the binary representation of a(n), reverse it, add 1 to each digit.  The result is the decimal representation of A102659(n).
%C A275692 From _Gus Wiseman_, Apr 19 2020: (Start)
%C A275692 Also numbers k such that the k-th composition in standard order (row k of A066099) is a Lyndon word. For example, the sequence of all Lyndon words begins:
%C A275692     0: ()            52: (1,2,3)        118: (1,1,2,1,2)
%C A275692     1: (1)           56: (1,1,4)        120: (1,1,1,4)
%C A275692     2: (2)           58: (1,1,2,2)      122: (1,1,1,2,2)
%C A275692     4: (3)           60: (1,1,1,3)      124: (1,1,1,1,3)
%C A275692     6: (1,2)         62: (1,1,1,1,2)    126: (1,1,1,1,1,2)
%C A275692     8: (4)           64: (7)            128: (8)
%C A275692    12: (1,3)         72: (3,4)          144: (3,5)
%C A275692    14: (1,1,2)       80: (2,5)          160: (2,6)
%C A275692    16: (5)           84: (2,2,3)        164: (2,3,3)
%C A275692    20: (2,3)         96: (1,6)          168: (2,2,4)
%C A275692    24: (1,4)         98: (1,4,2)        192: (1,7)
%C A275692    26: (1,2,2)      100: (1,3,3)        194: (1,5,2)
%C A275692    28: (1,1,3)      104: (1,2,4)        196: (1,4,3)
%C A275692    30: (1,1,1,2)    106: (1,2,2,2)      200: (1,3,4)
%C A275692    32: (6)          108: (1,2,1,3)      202: (1,3,2,2)
%C A275692    40: (2,4)        112: (1,1,5)        208: (1,2,5)
%C A275692    48: (1,5)        114: (1,1,3,2)      210: (1,2,3,2)
%C A275692    50: (1,3,2)      116: (1,1,2,3)      212: (1,2,2,3)
%C A275692 (End)
%H A275692 Robert Israel, <a href="/A275692/b275692.txt">Table of n, a(n) for n = 1..9868</a>
%e A275692 6 is in the sequence because its binary representation 110 is greater than all the rotations 011 and 101.
%e A275692 10 is not in the sequence because its binary representation 1010 is unchanged under rotation by 2 places.
%e A275692 From _Gus Wiseman_, Oct 31 2019: (Start)
%e A275692 The sequence of terms together with their binary expansions and binary indices begins:
%e A275692     1:       1 ~ {1}
%e A275692     2:      10 ~ {2}
%e A275692     4:     100 ~ {3}
%e A275692     6:     110 ~ {2,3}
%e A275692     8:    1000 ~ {4}
%e A275692    12:    1100 ~ {3,4}
%e A275692    14:    1110 ~ {2,3,4}
%e A275692    16:   10000 ~ {5}
%e A275692    20:   10100 ~ {3,5}
%e A275692    24:   11000 ~ {4,5}
%e A275692    26:   11010 ~ {2,4,5}
%e A275692    28:   11100 ~ {3,4,5}
%e A275692    30:   11110 ~ {2,3,4,5}
%e A275692    32:  100000 ~ {6}
%e A275692    40:  101000 ~ {4,6}
%e A275692    48:  110000 ~ {5,6}
%e A275692    50:  110010 ~ {2,5,6}
%e A275692    52:  110100 ~ {3,5,6}
%e A275692    56:  111000 ~ {4,5,6}
%e A275692    58:  111010 ~ {2,4,5,6}
%e A275692 (End)
%p A275692 filter:= proc(n) local L, k;
%p A275692   L:= convert(convert(n,binary),string);
%p A275692   for k from 1 to length(L)-1 do
%p A275692     if lexorder(L,StringTools:-Rotate(L,k)) then return false fi;
%p A275692   od;
%p A275692   true
%p A275692 end proc:
%p A275692 select(filter, [$0..1000]);
%t A275692 filterQ[n_] := Module[{bits, rr}, bits = IntegerDigits[n, 2]; rr = NestList[RotateRight, bits, Length[bits]-1] // Rest; AllTrue[rr, FromDigits[#, 2] < n&]];
%t A275692 Select[Range[0, 1000], filterQ] (* _Jean-François Alcover_, Apr 29 2019 *)
%o A275692 (Python)
%o A275692 def ok(n):
%o A275692     b = bin(n)[2:]
%o A275692     return all(b[i:] + b[:i] < b for i in range(1, len(b)))
%o A275692 print([k for k in range(230) if ok(k)]) # _Michael S. Branicky_, May 26 2022
%Y A275692 A similar concept is A328596.
%Y A275692 Numbers whose binary expansion is aperiodic are A328594.
%Y A275692 Numbers whose reversed binary expansion is a necklace are A328595.
%Y A275692 Binary necklaces are A000031.
%Y A275692 Binary Lyndon words are A001037.
%Y A275692 Lyndon compositions are A059966.
%Y A275692 Length of Lyndon factorization of binary expansion is A211100.
%Y A275692 Length of co-Lyndon factorization of binary expansion is A329312.
%Y A275692 Length of Lyndon factorization of reversed binary expansion is A329313.
%Y A275692 Length of co-Lyndon factorization of reversed binary expansion is A329326.
%Y A275692 Cf. A000031, A000740, A008965, A027375, A102659, A121016.
%Y A275692 All of the following pertain to compositions in standard order (A066099):
%Y A275692 - Length is A000120.
%Y A275692 - Necklaces are A065609.
%Y A275692 - Sum is A070939.
%Y A275692 - Rotational symmetries are counted by A138904.
%Y A275692 - Strict compositions are A233564.
%Y A275692 - Constant compositions are A272919.
%Y A275692 - Lyndon compositions are A275692 (this sequence).
%Y A275692 - Co-Lyndon compositions are A326774.
%Y A275692 - Rotational period is A333632.
%Y A275692 - Co-necklaces are A333764.
%Y A275692 - Co-Lyndon factorizations are counted by A333765.
%Y A275692 - Lyndon factorizations are counted by A333940.
%Y A275692 - Reversed necklaces are A333943.
%Y A275692 Cf. A034691, A060223, A124767, A269134, A292884.
%K A275692 nonn
%O A275692 1,3
%A A275692 _Robert Israel_, Aug 05 2016