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A328595
Numbers whose reversed binary expansion is a necklace.
44
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, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 136, 144, 152, 160, 164, 168, 170, 172, 176, 180
OFFSET
1,2
COMMENTS
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.
LINKS
EXAMPLE
The sequence of terms together with their binary expansions and binary indices begins:
1: 1 ~ {1}
2: 10 ~ {2}
3: 11 ~ {1,2}
4: 100 ~ {3}
6: 110 ~ {2,3}
7: 111 ~ {1,2,3}
8: 1000 ~ {4}
10: 1010 ~ {2,4}
12: 1100 ~ {3,4}
14: 1110 ~ {2,3,4}
15: 1111 ~ {1,2,3,4}
16: 10000 ~ {5}
20: 10100 ~ {3,5}
24: 11000 ~ {4,5}
26: 11010 ~ {2,4,5}
28: 11100 ~ {3,4,5}
30: 11110 ~ {2,3,4,5}
31: 11111 ~ {1,2,3,4,5}
32: 100000 ~ {6}
36: 100100 ~ {3,6}
MATHEMATICA
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Select[Range[100], neckQ[Reverse[IntegerDigits[#, 2]]]&]
PROG
(Python)
from itertools import count, islice
from sympy.utilities.iterables import necklaces
def a_gen():
for n in count(1):
t = []
for i in necklaces(n, 2):
if sum(i)>0:
t.append(sum(2**j for j in range(len(i)) if i[j] > 0))
yield from sorted(t)
A328595_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, May 24 2024
CROSSREFS
A similar concept is A065609.
The version with the most significant digit ignored is A328607.
Lyndon words are A328596.
Aperiodic words are A328594.
Binary necklaces are A000031.
Necklace compositions are A008965.
Sequence in context: A062974 A329399 A329396 * A065609 A334274 A225620
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Oct 22 2019
STATUS
approved