%I #16 May 24 2024 16:22:37
%S 1,2,3,4,6,7,8,10,12,14,15,16,20,24,26,28,30,31,32,36,40,42,44,48,52,
%T 54,56,58,60,62,63,64,72,80,84,88,92,96,100,104,106,108,112,116,118,
%U 120,122,124,126,127,128,136,144,152,160,164,168,170,172,176,180
%N Numbers whose reversed binary expansion is a necklace.
%C A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.
%H John Tyler Rascoe, <a href="/A328595/b328595.txt">Table of n, a(n) for n = 1..10000</a>
%e The sequence of terms together with their binary expansions and binary indices begins:
%e 1: 1 ~ {1}
%e 2: 10 ~ {2}
%e 3: 11 ~ {1,2}
%e 4: 100 ~ {3}
%e 6: 110 ~ {2,3}
%e 7: 111 ~ {1,2,3}
%e 8: 1000 ~ {4}
%e 10: 1010 ~ {2,4}
%e 12: 1100 ~ {3,4}
%e 14: 1110 ~ {2,3,4}
%e 15: 1111 ~ {1,2,3,4}
%e 16: 10000 ~ {5}
%e 20: 10100 ~ {3,5}
%e 24: 11000 ~ {4,5}
%e 26: 11010 ~ {2,4,5}
%e 28: 11100 ~ {3,4,5}
%e 30: 11110 ~ {2,3,4,5}
%e 31: 11111 ~ {1,2,3,4,5}
%e 32: 100000 ~ {6}
%e 36: 100100 ~ {3,6}
%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t Select[Range[100],neckQ[Reverse[IntegerDigits[#,2]]]&]
%o (Python)
%o from itertools import count, islice
%o from sympy.utilities.iterables import necklaces
%o def a_gen():
%o for n in count(1):
%o t = []
%o for i in necklaces(n,2):
%o if sum(i)>0:
%o t.append(sum(2**j for j in range(len(i)) if i[j] > 0))
%o yield from sorted(t)
%o A328595_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, May 24 2024
%Y A similar concept is A065609.
%Y The version with the most significant digit ignored is A328607.
%Y Lyndon words are A328596.
%Y Aperiodic words are A328594.
%Y Binary necklaces are A000031.
%Y Necklace compositions are A008965.
%Y Cf. A000120, A000740, A001037, A032153, A059966, A275692, A328668.
%K nonn,base
%O 1,2
%A _Gus Wiseman_, Oct 22 2019