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Rank of positive Josephus numbers A006257 in A000265 considered by 2^p terms.

In other words, numbers whose binary representation consists of one or more repeating blocks with only one 1 in each block.

Also, fixed points of the permutations A139706 and A139708.

Each a(n) is a term of A064896 multiplied by some power of 2. As such, this sequence must also be a subsequence of A125121.

Also the numbers that uniquely index a Haar graph (i.e., 5 and 6 are not in the sequence since H(5) is isomorphic to H(6)). - Eric W. Weisstein, Aug 19 2017

From Gus Wiseman, Apr 04 2020: (Start)

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. This sequence lists all positive integers k such that the k-th composition in standard order is constant. For example, the sequence together with the corresponding constant compositions begins:

0: () 136: (4,4)

1: (1) 170: (2,2,2,2)

2: (2) 255: (1,1,1,1,1,1,1,1)

3: (1,1) 256: (9)

4: (3) 292: (3,3,3)

7: (1,1,1) 511: (1,1,1,1,1,1,1,1,1)

8: (4) 512: (10)

10: (2,2) 528: (5,5)

15: (1,1,1,1) 682: (2,2,2,2,2)

16: (5) 1023: (1,1,1,1,1,1,1,1,1,1)

31: (1,1,1,1,1) 1024: (11)

32: (6) 2047: (1,1,1,1,1,1,1,1,1,1,1)

36: (3,3) 2048: (12)

42: (2,2,2) 2080: (6,6)

63: (1,1,1,1,1,1) 2184: (4,4,4)

64: (7) 2340: (3,3,3,3)

127: (1,1,1,1,1,1,1) 2730: (2,2,2,2,2,2)

128: (8) 4095: (1,1,1,1,1,1,1,1,1,1,1,1)

(End)

This sequence written in binary is A139707.

This is a permutation of the positive integers. A139708 is the inverse permutation.

Moreover, the first 2^n terms are a permutation of the first 2^n positive integers. Fixed points of the permutation are A272919. - Ivan Neretin, May 10 2016

This sequence written in decimal is A139708.

This sequence is a permutation of the nonnegative integers with inverse A343601.

This sequence can be extended to negative indexes by setting a(-n) = -a(n) for any n > 0. We then obtain a permutation of the integers (Z) with inverse A343601 (after a similar extension to negative indexes).

As a sequence, {a(n)} permutes the positive integers. As an array, {T(n,k)} is an interspersion-dispersion or I-D array (refer to Kimberling, 1st linked reference).

The top row of {T(n,k)} is A000079 or powers of two = 1, 2, 4, 8, 16, ....

Except for the leftmost element "1" of the top row, rows of T(n,k) indexed n = 0, 1, 2, ..., consist entirely of even numbers (A005843) for n even and entirely of odd numbers (A005408) for n odd.

The left column (k = 1) of {T(n,k)} comprises a "1" for the top row (n = 0) and A004755(n) = n + 2^(floor(log_2(n)) + 1), for rows n = 1, 2, 3, ....

For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., T(n,k) is given by T(0,k) = L^(k - 1)(1) and T(n,k) = L^(k - 1) R(n) for n = 1, 2, 3, ..., the image of n under a composition of branching functions L(n) = A004754(n) = n + 2^floor(log_2(n)) and R(n) = A004755(n) = n + 2^(floor(log_2(n)) + 1) (cf. generating tree A059893 and 2nd linked reference).

(Duality with array A054582): Consider A059893 and A000027 as labeled binary trees arranging the positive integers. In latter tree, node labels equal node positions, thus following their natural order. Rows of {T(n,k)} are the labels along maximal straight paths that always branch left in the former tree, while rows of (transposed) array A054582 are the labels along maximal straight paths that always branch left in the latter tree.

Column k of {T(n,k)} comprises the (sorted) labels in the k-th right clade of latter tree, while column k of (transposed) A054582 comprises the (sorted) labels in the k-th right clade of the former tree. This makes the arrays {T(n,k)} and (transposed) A054582 "blade-duals," blade being a contraction of branch-clade ('right clades' explained under tree A345253 and in 2nd link).

Write the positive integers in natural order as a (left-justified) "tetrangle" or "irregular triangle" tableau with 2^t entries on each row t, for t=1, 2, 3, .... Then, columns of the tableau equal rows of {T(n,k)} (2nd link):

1,

2, 3,

4, 5, 6, 7,

8, 9, 10, 11, 12, 13, 14, 15

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,

...

Analogous to A345252, its right-justified tableau of the positive integers in cohorts with lengths the Fibonacci numbers replaced by the above left-justified tableau with powers of two as lengths of the cohorts.

(Mirror duality): A "mirror dual" I-D array or "inverse I-D array" (see Kimberling, 1st linked reference) is obtained by substituting the left-justified tableau by a right-justified tableau and following the identical procedure, or equivalently by mirroring the tree A059893 cited above, i.e., taking maximal straight paths that always branch right in the tree A059893. With two types of duality then, {T(n,k)} forms a quartet of I-D arrays together with its mirror dual, its blade dual (transposed) A054582 and mirror dual A191448 of the latter.

(Para-sequences): Sequences of row and column indices (see Conway-Sloane correspondence under A019586, citing Kimberling). For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., the row index n of positive integer T(n,k) is A053645(T) and the column index k of positive integer T(n,k) is A065120(T).