A328595 Numbers whose reversed binary expansion is a necklace.

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

Author

Gus Wiseman, Oct 22 2019

Keywords

• nonn
• base

Comments

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

Examples

			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}
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..10000

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.

Cf. A000120, A000740, A001037, A032153, A059966, A275692, A328668.

Programs

Mathematica

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],neckQ[Reverse[IntegerDigits[#,2]]]&]

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