A065609 -id:A065609 - cp's OEIS frontend

A225620 Indices of partitions in the table of compositions of A228351.

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 100, 104, 106, 112, 116, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 200, 208, 212, 224, 228, 232, 234, 240, 244, 248, 250, 252, 254, 255

Offset: 1

Omar E. Pol, Aug 03 2013

Also triangle read by rows in which T(n,k) is the decimal representation of a binary number whose mirror represents the k-th partition of n according with the list of juxtaposed reverse-lexicographically ordered partitions of the positive integers (A026792).
In order to construct this sequence as a triangle we use the following rules:
- In the list of A026792 we replace each part of size j of the k-th partition of n by concatenation of j - 1 zeros and only one 1.
- Then replace this new set of parts by the concatenation of its parts.
- Then replace this string by its mirror version which is a binary number.
T(n,k) is the decimal value of this binary number, which represents the k-th partition of n (see example).
The partitions of n are represented by a subsequence with A000041(n) integers starting with 2^(n-1) and ending with 2^n - 1, n >= 1. The odd numbers of the sequence are in A000225.
First differs from A065609 at a(23).
Conjecture: this sequence is a sorted version of b(n) where b(2^k) = 2^k for k >= 0, b(n) = A080100(n)*(2*b(A053645(n)) + 1) otherwise. - Mikhail Kurkov, Oct 21 2023

			T(6,8) = 58 because 58 in base 2 is 111010 whose mirror is 010111 which is the concatenation of 01, 01, 1, 1, whose number of digits are 2, 2, 1, 1, which are also the 8th partition of 6.
Illustration of initial terms:
The sequence represents a table of partitions (see below):
--------------------------------------------------------
.            Binary                        Partitions
n  k  T(n,k) number  Mirror   Diagram       (A026792)
.                                          1 2 3 4 5 6
--------------------------------------------------------
.                             _
1  1     1       1    1        |           1,
.                             _ _
1  1     2      10    01      _  |           2,
2  2     3      11    11       | |         1,1,
.                             _ _ _
3  1     4     100    001     _ _  |           3,
3  2     6     110    011     _  | |         2,1,
3  3     7     111    111      | | |       1,1,1,
.                             _ _ _ _
4  1     8    1000    0001    _ _    |           4,
4  2    10    1010    0101    _ _|_  |         2,2,
4  3    12    1100    0011    _ _  | |         3,1,
4  4    14    1110    0111    _  | | |       2,1,1,
4  5    15    1111    1111     | | | |     1,1,1,1,
.                             _ _ _ _ _
5  1    16   10000    00001   _ _ _    |           5,
5  2    20   10100    00101   _ _ _|_  |         3,2,
5  3    24   11000    00011   _ _    | |         4,1,
5  4    26   11010    01011   _ _|_  | |       2,2,1,
5  5    28   11100    00111   _ _  | | |       3,1,1,
5  6    30   11110    01111   _  | | | |     2,1,1,1,
5  7    31   11111    11111    | | | | |   1,1,1,1,1,
.                             _ _ _ _ _ _
6  1    32  100000    000001  _ _ _      |           6
6  2    36  100100    001001  _ _ _|_    |         3,3,
6  3    40  101000    000101  _ _    |   |         4,2,
6  4    42  101010    010101  _ _|_ _|_  |       2,2,2,
6  5    48  110000    000011  _ _ _    | |         5,1,
6  6    52  110100    001011  _ _ _|_  | |       3,2,1,
6  7    56  111000    000111  _ _    | | |       4,1,1,
6  8    58  111010    010111  _ _|_  | | |     2,2,1,1,
6  9    60  111100    001111  _ _  | | | |     3,1,1,1,
6  10   62  111110    011111  _  | | | | |   2,1,1,1,1,
6  11   63  111111    111111   | | | | | | 1,1,1,1,1,1,
.
Triangle begins:
  1;
  2,   3;
  4,   6,  7;
  8,  10, 12, 14, 15;
  16, 20, 24, 26, 28, 30, 31;
  32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63;
  ...
From _Gus Wiseman_, Apr 01 2020: (Start)
Using the encoding of A066099, this sequence ranks all finite nonempty multisets, as follows.
   1: {1}
   2: {2}
   3: {1,1}
   4: {3}
   6: {1,2}
   7: {1,1,1}
   8: {4}
  10: {2,2}
  12: {1,3}
  14: {1,1,2}
  15: {1,1,1,1}
  16: {5}
  20: {2,3}
  24: {1,4}
  26: {1,2,2}
  28: {1,1,3}
  30: {1,1,1,2}
  31: {1,1,1,1,1}
(End)

Column 1 is A000079. Row n has length A000041(n). Right border gives A000225.
Cf. A026792, A065609, A135010, A141285, A186114, A194446, A194546, A206437, A207779, A211978, A225600, A225610, A228351.
The case covering an initial interval is A333379 or A333380.
All of the following pertain to compositions in the order of A066099.
- The weakly increasing version is this sequence.
- The weakly decreasing version is A114994.
- The strictly increasing version is A333255.
- The strictly decreasing version is A333256.
- The unequal version is A233564.
- The equal version is A272919.
- The case covering an initial interval is A333217.
- Initial intervals are ranked by A164894.
- Reversed initial intervals are ranked by A246534.
Cf. A000120, A029931, A048793, A070939, A124766, A233249, A333219, A333220.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],LessEqual@@stc[#]&] (* Gus Wiseman, Apr 01 2020 *)

Conjecture: a(A000070(m) - k) = 2^m - A228354(k) for m > 0, 0 < k <= A000041(m). - Mikhail Kurkov, Oct 20 2023

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

Original entry on oeis.org

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

A328594 Numbers whose binary expansion is aperiodic.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Oct 22 2019

nonn

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

			The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}

The complement is A121016.
The version for prime indices is A085971.
Numbers without proper integer roots are A007916.
Necklaces are A328595.
Lyndon words are A328596.
Aperiodic compositions are A000740.
Aperiodic binary sequences are A027375.
Cf. A000120, A000939, A014081, A065609, A069010, A275692, A323867, A334030.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Select[Range[0,100],aperQ[IntegerDigits[#,2]]&]

A328595 Numbers whose reversed binary expansion is a necklace.

Original entry on oeis.org

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

Gus Wiseman, Oct 22 2019

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

			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}

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

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.

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

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

A334028 Number of distinct parts in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2

Offset: 0

Gus Wiseman, Apr 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 77th composition is (3,1,2,1), so a(77) = 3.

Number of distinct prime indices is A001221.
Positions of first appearances (offset 1) are A246534.
Positions of 1's are A272919.
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.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Dealings are A333939.
Cf. A001037, A059966, A060223, A066099, A333765, A333940.

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

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

A121016 Numbers whose binary expansion is properly periodic.

Original entry on oeis.org

3, 7, 10, 15, 31, 36, 42, 45, 54, 63, 127, 136, 153, 170, 187, 204, 221, 238, 255, 292, 365, 438, 511, 528, 561, 594, 627, 660, 682, 693, 726, 759, 792, 825, 858, 891, 924, 957, 990, 1023, 2047, 2080, 2145, 2184, 2210, 2275, 2340, 2405, 2457, 2470, 2535

Offset: 1

Jacob A. Siehler, Sep 08 2006

A finite sequence is aperiodic if its cyclic rotations are all different. - Gus Wiseman, Oct 31 2019

			For example, 204=(1100 1100)_2 and 292=(100 100 100)_2 belong to the sequence, but 30=(11110)_2 cannot be split into repeating periods.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   3:         11 ~ {1,2}
   7:        111 ~ {1,2,3}
   10:      1010 ~ {2,4}
   15:      1111 ~ {1,2,3,4}
   31:     11111 ~ {1,2,3,4,5}
   36:    100100 ~ {3,6}
   42:    101010 ~ {2,4,6}
   45:    101101 ~ {1,3,4,6}
   54:    110110 ~ {2,3,5,6}
   63:    111111 ~ {1,2,3,4,5,6}
  127:   1111111 ~ {1,2,3,4,5,6,7}
  136:  10001000 ~ {4,8}
  153:  10011001 ~ {1,4,5,8}
  170:  10101010 ~ {2,4,6,8}
  187:  10111011 ~ {1,2,4,5,6,8}
  204:  11001100 ~ {3,4,7,8}
  221:  11011101 ~ {1,3,4,5,7,8}
  238:  11101110 ~ {2,3,4,6,7,8}
  255:  11111111 ~ {1,2,3,4,5,6,7,8}
  292: 100100100 ~ {3,6,9}
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A020330 is a subsequence.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary indices have equal run-lengths are A164707.
Cf. A000120, A003714, A014081, A065609, A069010, A275692, A328595.

PeriodicQ[n_, base_] := Block[{l = IntegerDigits[n, base]}, MemberQ[ RotateLeft[l, # ] & /@ Most@ Divisors@ Length@l, l]]; Select[ Range@2599, PeriodicQ[ #, 2] &]

is(n)=n=binary(n);fordiv(#n,d,for(i=1,#n/d-1, for(j=1,d, if(n[j]!=n[j+i*d], next(3)))); return(d<#n)) \\ Charles R Greathouse IV, Dec 10 2013

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
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See Links section.
```

A333764 Numbers k such that the k-th composition in standard order is a co-necklace.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140

Offset: 1

Gus Wiseman, Apr 12 2020

nonn

A co-necklace is a finite sequence that is lexicographically greater than or equal to any cyclic rotation.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 sequence together with the corresponding co-necklaces begins:
    1: (1)             32: (6)               69: (4,2,1)
    2: (2)             33: (5,1)             70: (4,1,2)
    3: (1,1)           34: (4,2)             71: (4,1,1,1)
    4: (3)             35: (4,1,1)           73: (3,3,1)
    5: (2,1)           36: (3,3)             74: (3,2,2)
    7: (1,1,1)         37: (3,2,1)           75: (3,2,1,1)
    8: (4)             38: (3,1,2)           77: (3,1,2,1)
    9: (3,1)           39: (3,1,1,1)         78: (3,1,1,2)
   10: (2,2)           42: (2,2,2)           79: (3,1,1,1,1)
   11: (2,1,1)         43: (2,2,1,1)         85: (2,2,2,1)
   15: (1,1,1,1)       45: (2,1,2,1)         87: (2,2,1,1,1)
   16: (5)             47: (2,1,1,1,1)       91: (2,1,2,1,1)
   17: (4,1)           63: (1,1,1,1,1,1)     95: (2,1,1,1,1,1)
   18: (3,2)           64: (7)              127: (1,1,1,1,1,1,1)
   19: (3,1,1)         65: (6,1)            128: (8)
   21: (2,2,1)         66: (5,2)            129: (7,1)
   23: (2,1,1,1)       67: (5,1,1)          130: (6,2)
   31: (1,1,1,1,1)     68: (4,3)            131: (6,1,1)

The non-"co" version is A065609.
The reversed version is A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Necklaces covering an initial interval are A019536.
Numbers whose prime signature is a necklace are A329138.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- 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.
- Aperiodic compositions are A328594.
- Length of Lyndon factorization is A329312.
- Rotational period is A333632.
- Reversed necklaces are A333943.
Cf. A000740, A001037, A027375, A059966, A211100, A302291, A328596, A329142, A333765, A333939, A333941.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
coneckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[100],coneckQ[stc[#]]&]

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).

cp's OEIS Frontend

A225620 Indices of partitions in the table of compositions of A228351.

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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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A328595 Numbers whose reversed binary expansion is a necklace.

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A334028 Number of distinct parts in the n-th composition in standard order.

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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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A121016 Numbers whose binary expansion is properly periodic.

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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.

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