A121016 -id:A121016 - cp's OEIS frontend

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

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, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 128, 144, 160, 164, 168, 192, 194, 196, 200, 202, 208, 210, 212, 216, 218, 224, 226, 228

Offset: 1

Robert Israel, Aug 05 2016

nonn

0, and terms of A065609 that are not in A121016.
Number of terms with d binary digits is A001037(d).
Take the binary representation of a(n), reverse it, add 1 to each digit.  The result is the decimal representation of A102659(n).
From Gus Wiseman, Apr 19 2020: (Start)
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:
()            52: (1,2,3)        118: (1,1,2,1,2)
(1)           56: (1,1,4)        120: (1,1,1,4)
(2)           58: (1,1,2,2)      122: (1,1,1,2,2)
(3)           60: (1,1,1,3)      124: (1,1,1,1,3)
(1,2)         62: (1,1,1,1,2)    126: (1,1,1,1,1,2)
(4)           64: (7)            128: (8)
(1,3)         72: (3,4)          144: (3,5)
(1,1,2)       80: (2,5)          160: (2,6)
(5)           84: (2,2,3)        164: (2,3,3)
(2,3)         96: (1,6)          168: (2,2,4)
(1,4)         98: (1,4,2)        192: (1,7)
(1,2,2)      100: (1,3,3)        194: (1,5,2)
(1,1,3)      104: (1,2,4)        196: (1,4,3)
(1,1,1,2)    106: (1,2,2,2)      200: (1,3,4)
(6)          108: (1,2,1,3)      202: (1,3,2,2)
(2,4)        112: (1,1,5)        208: (1,2,5)
(1,5)        114: (1,1,3,2)      210: (1,2,3,2)
(1,3,2)      116: (1,1,2,3)      212: (1,2,2,3)
(End)

			6 is in the sequence because its binary representation 110 is greater than all the rotations 011 and 101.
10 is not in the sequence because its binary representation 1010 is unchanged under rotation by 2 places.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    6:     110 ~ {2,3}
    8:    1000 ~ {4}
   12:    1100 ~ {3,4}
   14:    1110 ~ {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}
   32:  100000 ~ {6}
   40:  101000 ~ {4,6}
   48:  110000 ~ {5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   56:  111000 ~ {4,5,6}
   58:  111010 ~ {2,4,5,6}
(End)

Robert Israel, Table of n, a(n) for n = 1..9868

A similar concept is A328596.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is a necklace are A328595.
Binary necklaces are A000031.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Length of Lyndon factorization of binary expansion is A211100.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
Cf. A000031, A000740, A008965, A027375, A102659, A121016.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692 (this sequence).
- Co-Lyndon compositions are A326774.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
Cf. A034691, A060223, A124767, A269134, A292884.

filter:= proc(n) local L, k;
  L:= convert(convert(n,binary),string);
  for k from 1 to length(L)-1 do
    if lexorder(L,StringTools:-Rotate(L,k)) then return false fi;
  od;
  true
end proc:
select(filter, [$0..1000]);

filterQ[n_] := Module[{bits, rr}, bits = IntegerDigits[n, 2]; rr = NestList[RotateRight, bits, Length[bits]-1] // Rest; AllTrue[rr, FromDigits[#, 2] < n&]];
Select[Range[0, 1000], filterQ] (* Jean-François Alcover, Apr 29 2019 *)

def ok(n):
    b = bin(n)[2:]
    return all(b[i:] + b[:i] < b for i in range(1, len(b)))
print([k for k in range(230) if ok(k)]) # Michael S. Branicky, May 26 2022

A178472 Number of compositions (ordered partitions) of n where the gcd of the part sizes is not 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 1, 32768, 1027, 65537, 79, 133088, 1, 262145, 4099, 524408, 1, 1056731, 1, 2097158, 16636, 4194305, 1, 8421248, 64, 16777712, 65539

Offset: 1

Franklin T. Adams-Watters, May 28 2010

nonn

Of course, all part sizes must be greater than 1; that condition alone gives the Fibonacci numbers, which is thus an upper bound.

Also the number of periodic compositions of n, where a sequence is periodic if its cyclic rotations are not all different. Also compositions with non-relatively prime run-lengths. - Gus Wiseman, Nov 10 2019

			For n=6, we have 5 compositions: <6>, <4,2>, <2,4>, <2,2,2>, and <3,3>.
From _Gus Wiseman_, Nov 10 2019: (Start)
The a(2) = 1 through a(9) = 4 non-relatively prime compositions:
  (2)  (3)  (4)    (5)  (6)      (7)  (8)        (9)
            (2,2)       (2,4)         (2,6)      (3,6)
                        (3,3)         (4,4)      (6,3)
                        (4,2)         (6,2)      (3,3,3)
                        (2,2,2)       (2,2,4)
                                      (2,4,2)
                                      (4,2,2)
                                      (2,2,2,2)
The a(2) = 1 through a(9) = 4 periodic compositions:
  11  111  22    11111  33      1111111  44        333
           1111         222              1313      121212
                        1212             2222      212121
                        2121             3131      111111111
                        111111           112112
                                         121121
                                         211211
                                         11111111
The a(2) = 1 through a(9) = 4 compositions with non-relatively prime run-lengths:
  11  111  22    11111  33      1111111  44        333
           1111         222              1133      111222
                        1122             2222      222111
                        2211             3311      111111111
                        111111           111122
                                         112211
                                         221111
                                         11111111
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014. See Table 2.

Cf. A000045, A008683, A011782, A178470.
Periodic binary words are A152061.
Cf. A000740, A027375, A059966, A121016, A329140, A329145.

A178472 := n -> (2^n - add(mobius(n/d)*2^d, d in divisors(n)))/2:
seq(A178472(n), n=1..51); # Peter Luschny, Jan 21 2018

Table[2^(n - 1) - DivisorSum[n, MoebiusMu[n/#]*2^(# - 1) &], {n, 51}] (* Michael De Vlieger, Jan 20 2018 *)

vector(60,n,2^(n-1)-sumdiv(n,d,2^(d-1)*moebius(n/d)))

from sympy import mobius, divisors
def A178472(n): return -sum(mobius(n//d)<Chai Wah Wu, Sep 21 2024

a(n) = Sum_{d|n & d

a(n) = 2^(n-1) - A000740(n).

a(n) = A152061(n)/2. - George Beck, Jan 20 2018

a(p) = 1 for p prime. - Chai Wah Wu, Sep 21 2024

Ambiguous term a(0) removed by Max Alekseyev, Jan 02 2012

A020330 Numbers whose base-2 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

3, 10, 15, 36, 45, 54, 63, 136, 153, 170, 187, 204, 221, 238, 255, 528, 561, 594, 627, 660, 693, 726, 759, 792, 825, 858, 891, 924, 957, 990, 1023, 2080, 2145, 2210, 2275, 2340, 2405, 2470, 2535, 2600, 2665, 2730, 2795, 2860, 2925, 2990, 3055, 3120, 3185, 3250

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

All differences are in union of A000051 and A001576. - Vladimir Shevelev, Dec 07 2013

			36 is a term because 36 = 100100_2, which is 100 followed by 100.

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191
Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's Theorem for Binary Powers, Combinatorica, Vol. 39, No. 6 (2019), pp. 1335-1350, arXiv preprint, arXiv:1801.04483 [math.NT], 2018.
Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, arXiv:1710.04247 [math.NT], 2017-2018.
Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatshefte für Mathematik, Vol. 161, No. 1 (2010), pp. 85-114, alternative link.
Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.

Subsequence of A121016.
Cf. A000051, A001576, A007088, A030308, A062383, A070939, A330157.
Column k=0 of A246830, column k=1 of A246834.

a020330 n = foldr (\d v -> 2 * v + d) 0 (bs ++ bs) where
   bs = a030308_row n
-- Reinhard Zumkeller, Feb 19 2013

[n+2*n*2^Floor(Log(2, n)): n in [1..50]]; // Vincenzo Librandi, Apr 05 2018

a:= n-> (l-> Bits[Join]([l[],l[]]))(Bits[Split](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 24 2024

Table[n + 2 n 2^Floor[Log[2, n]], {n, 50}] (* T. D. Noe, Dec 10 2013 *)
FromDigits[#, 2] & /@ (# <> # & /@ IntegerString[Range@100, 2]) (* Hans Rudolf Widmer, Aug 24 2024 *)

a(n)=n+n<<#binary(n) \\ Charles R Greathouse IV, Mar 29 2013

is(n)=my(L=#binary(n)\2); n>>L==bitand(n,2^L-1) \\ Charles R Greathouse IV, Mar 29 2013

def a(n): return int(bin(n)[2:]*2, 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 10 2021

def A020330(n): return (n<Chai Wah Wu, Feb 28 2023

a(n) = n + 2*n*2^floor(log_2(n)). - Ralf Stephan, Dec 07 2004
Sum_{n>=1} 1/a(n) = A330157. - Amiram Eldar, Oct 22 2020
a(n) = n * (2^A070939(n) + 1). - Jianing Song, Apr 10 2021

A164707 A positive integer n is included if all runs of 1's in binary n are of the same length.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 27, 28, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 48, 51, 54, 56, 60, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 96, 99, 102, 108, 112, 119, 120, 124, 126, 127, 128, 129, 130, 132, 133, 136

Offset: 1

Leroy Quet, Aug 23 2009

Clarification: A binary number consists of "runs" completely of 1's alternating with runs completely of 0's. No two or more runs all of the same digit are adjacent.

This sequence contains in part positive integers that each contain one run of 1's. For those members of this sequence each with at least two runs of 1's, see A164709.

			From _Gus Wiseman_, Oct 31 2019: (Start)
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}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
   8:   1000 ~ {4}
   9:   1001 ~ {1,4}
  10:   1010 ~ {2,4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  16:  10000 ~ {5}
  17:  10001 ~ {1,5}
  18:  10010 ~ {2,5}
  20:  10100 ~ {3,5}
  21:  10101 ~ {1,3,5}
  24:  11000 ~ {4,5}
  27:  11011 ~ {1,2,4,5}
(End)

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A164708, A164709, A164710.
The version for prime indices is A072774.
The binary expansion of n has A069010(n) runs of 1's.
Numbers whose runs are all of different lengths are A328592.
Partitions with equal multiplicities are A047966.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Cf. A000120, A003714, A014081, A065609, A070939, A121016, A245563, A275692.

isA164707 := proc(n) local bdg,arl,lset ; bdg := convert(n,base,2) ; lset := {} ; arl := -1 ; for p from 1 to nops(bdg) do if op(p,bdg) = 1 then if p = 1 then arl := 1 ; else arl := arl+1 ; end if; else if arl > 0 then lset := lset union {arl} ; end if; arl := 0 ; end if; end do ; if arl > 0 then lset := lset union {arl} ; end if; return (nops(lset) <= 1 ); end proc: for n from 1 to 300 do if isA164707(n) then printf("%d,",n) ; end if; end do; # R. J. Mathar, Feb 27 2010

Select[Range@ 140, SameQ @@ Map[Length, Select[Split@ IntegerDigits[#, 2], First@ # == 1 &]] &] (* Michael De Vlieger, Aug 20 2017 *)

foreach(1..140){
    %runs=();
    $runs{$}++ foreach split /0+/, sprintf("%b",$);
    print "$_, " if 1==keys(%runs);
}
# Ivan Neretin, Nov 09 2015

Extended beyond 42 by R. J. Mathar, Feb 27 2010

A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 1, 3, 2, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3

Offset: 0

Gus Wiseman, Nov 11 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

			The sequence of reversed binary expansions of the nonnegative integers together with their Lyndon factorizations begins:
   0:      () = ()
   1:     (1) = (1)
   2:    (01) = (01)
   3:    (11) = (1)(1)
   4:   (001) = (001)
   5:   (101) = (1)(01)
   6:   (011) = (011)
   7:   (111) = (1)(1)(1)
   8:  (0001) = (0001)
   9:  (1001) = (1)(001)
  10:  (0101) = (01)(01)
  11:  (1101) = (1)(1)(01)
  12:  (0011) = (0011)
  13:  (1011) = (1)(011)
  14:  (0111) = (0111)
  15:  (1111) = (1)(1)(1)(1)
  16: (00001) = (00001)
  17: (10001) = (1)(0001)
  18: (01001) = (01)(001)
  19: (11001) = (1)(1)(001)
  20: (00101) = (00101)

The non-reversed version is A211100.
Positions of 1's are A328596.
The "co" version is A329326.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose reversed binary expansion is a aperiodic are A328594.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A027375, A059966, A060223, A121016, A211097, A275692, A329131, A329314,  A329317, A329325.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
Table[If[n==0,0,Length[lynfac[Reverse[IntegerDigits[n,2]]]]],{n,0,30}]

A326774 For any number m, let m* be the bi-infinite string obtained by repetition of the binary representation of m; this sequence lists the numbers n such that for any k < n, n* does not equal k* up to a shift.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 38, 39, 43, 47, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 146, 147, 149, 150, 151, 154

Offset: 0

Rémy Sigrist, Jul 27 2019

This sequence contains every power of 2.
No term belongs to A121016.
Every terms belongs to A004761.
For any k > 0, there are A001037(k) terms with binary length k.
From Gus Wiseman, Apr 19 2020: (Start)
Also numbers k such that the k-th composition in standard order (row k of A066099) is a co-Lyndon word (regular Lyndon words being A275692). For example, the sequence of all co-Lyndon words begins:
()            37: (3,2,1)         79: (3,1,1,1,1)
(1)           38: (3,1,2)         85: (2,2,2,1)
(2)           39: (3,1,1,1)       87: (2,2,1,1,1)
(3)           43: (2,2,1,1)       91: (2,1,2,1,1)
(2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
(4)           64: (7)            128: (8)
(3,1)         65: (6,1)          129: (7,1)
(2,1,1)       66: (5,2)          130: (6,2)
(5)           67: (5,1,1)        131: (6,1,1)
(4,1)         68: (4,3)          132: (5,3)
(3,2)         69: (4,2,1)        133: (5,2,1)
(3,1,1)       70: (4,1,2)        134: (5,1,2)
(2,2,1)       71: (4,1,1,1)      135: (5,1,1,1)
(2,1,1,1)     73: (3,3,1)        137: (4,3,1)
(6)           74: (3,2,2)        138: (4,2,2)
(5,1)         75: (3,2,1,1)      139: (4,2,1,1)
(4,2)         77: (3,1,2,1)      140: (4,1,3)
(4,1,1)       78: (3,1,1,2)      141: (4,1,2,1)
(End)

			3* = ...11... equals 1* = ...1..., so 3 is not a term.
6* = ...110... equals up to a shift 5* = ...101..., so 6 is not a term.
11* = ...1011... only equals up to a shift 13* = ...1101... and 14* = ...1110..., so 11 is a term.

Rémy Sigrist, Logarithmic scatterplot of the first differences of the terms < 2^24
Rémy Sigrist, PARI program for A326774

Cf. A001037, A004761, A065609, A121016.
Necklace compositions are counted by A008965.
Lyndon compositions are counted by A059966.
Length of Lyndon factorization of binary expansion is A211100.
Numbers whose reversed binary expansion is a necklace are A328595.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774 (this sequence).
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
- Length of co-Lyndon factorization is A334029.
Cf. A000740, A034691, A060223, A269134, A292884, A328596.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],colynQ[stc[#]]&] (* Gus Wiseman, Apr 19 2020 *)

PARI
```
See Links section.
```

A333632 Rotational period of the k-th composition in standard order; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 1, 2, 2, 3, 1, 3, 3, 4, 2, 3, 1, 4, 3, 2, 4, 5, 2, 3, 3, 4, 3, 4, 2, 5, 3, 4, 4, 5, 4, 5, 5, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4

Offset: 0

Gus Wiseman, Apr 12 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The a(299) = 5 rotations:
  (1,1,3,2,2)
  (1,3,2,2,1)
  (3,2,2,1,1)
  (2,2,1,1,3)
  (2,1,1,3,2)
The a(9933) = 4 rotations:
  (1,2,1,3,1,2,1,3)
  (1,3,1,2,1,3,1,2)
  (2,1,3,1,2,1,3,1)
  (3,1,2,1,3,1,2,1)

Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
The version for binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
Compositions by number of distinct rotations are A333941.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Equal runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632 (this sequence).
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Array[RotateRight[stc[n],#]&,DigitCount[n,2,1]]]],{n,0,100}]

a(n) = A000120(n)/A138904(n) = A302291(n) - A023416(n)/A138904(n).

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A finite sequence is aperiodic if all of its cyclic rotations are distinct. See A000740 or A027375 for details.

Also numbers k such that the k-th composition in standard order is aperiodic. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 28 2020

Also numbers whose binary indices have different lengths of runs of successive parts. A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The complement is {5, 9, 10, 17, 18, 20, 21, 27, ...}.

cp's OEIS Frontend

A328596 Numbers whose reversed binary expansion is a Lyndon word (aperiodic necklace).

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

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A328594 Numbers whose binary expansion is aperiodic.

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A328592 Numbers whose binary expansion has all different lengths of runs of 1's.

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A178472 Number of compositions (ordered partitions) of n where the gcd of the part sizes is not 1.

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A020330 Numbers whose base-2 representation is the juxtaposition of two identical strings.

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A164707 A positive integer n is included if all runs of 1's in binary n are of the same length.

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