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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:

0: () 52: (1,2,3) 118: (1,1,2,1,2)

1: (1) 56: (1,1,4) 120: (1,1,1,4)

2: (2) 58: (1,1,2,2) 122: (1,1,1,2,2)

4: (3) 60: (1,1,1,3) 124: (1,1,1,1,3)

6: (1,2) 62: (1,1,1,1,2) 126: (1,1,1,1,1,2)

8: (4) 64: (7) 128: (8)

12: (1,3) 72: (3,4) 144: (3,5)

14: (1,1,2) 80: (2,5) 160: (2,6)

16: (5) 84: (2,2,3) 164: (2,3,3)

20: (2,3) 96: (1,6) 168: (2,2,4)

24: (1,4) 98: (1,4,2) 192: (1,7)

26: (1,2,2) 100: (1,3,3) 194: (1,5,2)

28: (1,1,3) 104: (1,2,4) 196: (1,4,3)

30: (1,1,1,2) 106: (1,2,2,2) 200: (1,3,4)

32: (6) 108: (1,2,1,3) 202: (1,3,2,2)

40: (2,4) 112: (1,1,5) 208: (1,2,5)

48: (1,5) 114: (1,1,3,2) 210: (1,2,3,2)

50: (1,3,2) 116: (1,1,2,3) 212: (1,2,2,3)

(End)

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:

0: () 37: (3,2,1) 79: (3,1,1,1,1)

1: (1) 38: (3,1,2) 85: (2,2,2,1)

2: (2) 39: (3,1,1,1) 87: (2,2,1,1,1)

4: (3) 43: (2,2,1,1) 91: (2,1,2,1,1)

5: (2,1) 47: (2,1,1,1,1) 95: (2,1,1,1,1,1)

8: (4) 64: (7) 128: (8)

9: (3,1) 65: (6,1) 129: (7,1)

11: (2,1,1) 66: (5,2) 130: (6,2)

16: (5) 67: (5,1,1) 131: (6,1,1)

17: (4,1) 68: (4,3) 132: (5,3)

18: (3,2) 69: (4,2,1) 133: (5,2,1)

19: (3,1,1) 70: (4,1,2) 134: (5,1,2)

21: (2,2,1) 71: (4,1,1,1) 135: (5,1,1,1)

23: (2,1,1,1) 73: (3,3,1) 137: (4,3,1)

32: (6) 74: (3,2,2) 138: (4,2,2)

33: (5,1) 75: (3,2,1,1) 139: (4,2,1,1)

34: (4,2) 77: (3,1,2,1) 140: (4,1,3)

35: (4,1,1) 78: (3,1,1,2) 141: (4,1,2,1)

(End)

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

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.

Mersenne numbers of form (2^n - 1) have n rotational symmetries.

For prime length binary expansions these are the only nontrivial symmetries.

For composite length expansions it seems that when the number of symmetries is nontrivial it is equal to a factor of the length. We're working on an explicit formula.

Discovered in the context of random circulant matrices, examining if there's a correlation between degrees of freedom and number of symmetries in the first row.

When combined with A138954, these two sequences should give a full account of the number of redundant rows in a circulant square matrix with at most two distinct values, where a(n) is the encoding of the first row of the matrix into binary such that value a = 1 and value b = 0.

Discovered on the night of Apr 02, 2008 by Maxwell Sills and Gary Doran.

Conjecture: For binary expansions of length n, there are d(n) distinct values that will show up as symmetries, where d is the divisor function. The symmetry values will be precisely the divisors of n.

Example: for binary expansions of length 12, one sees that d(12) = 6 distinct values show up as symmetries (1, 2, 3, 4, 6, 12).

Conjecture: For numbers whose binary expansion has length n which has proper divisors which are all coprime: There will be only one number of length n with n symmetries. That number is 2^n - 1. For each proper divisor d (excluding 1), you can generate all numbers of length n that have n/d symmetries like so: (2^0 + 2^d + 2^2d ... 2^(n-d)) * a, where 2^(d-1) <= a < (2^d) - 1. The rest of the expansions of length n will have only the trivial symmetry.

Also the number of rotational symmetries of the n-th composition in standard order (graded reverse-lexicographic). This composition (row n of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. - Gus Wiseman, Apr 19 2020

From Gus Wiseman, Apr 19 2020: (Start)

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.

Period of binary expansion is A302291.

Compositions by sum and 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.

- Runs are counted by A124767.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Lyndon compositions are A275692.

- Co-Lyndon compositions are A326774.

- Aperiodic compositions are A328594.

- Reversed co-necklaces are A328595.

- Rotational period is A333632.

- Co-necklaces are A333764.

- Reversed necklaces are A333943.

Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.

(End).

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 factorization of a composition c is a multiset of compositions whose Lyndon product is c.

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.

Also the number of multiset partitions of the Lyndon-word factorization of the n-th composition in standard order.

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon factorization of a composition c is a multiset of compositions whose co-Lyndon product is c.

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.

Also the number of multiset partitions of the co-Lyndon-word factorization of the n-th composition in standard order.

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.

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.

Reversed Lyndon words are different from co-Lyndon words (A326774).

A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations.

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.

Zero is assumed to be represented as 0; otherwise, leading zeros are ignored.

See A302295 for the variant where leading zeros are allowed.