A137706 -id:A137706 - cp's OEIS frontend

A272919 Numbers of the form 2^(n-1)(2^(nm)-1)/(2^n-1), n >= 1, m >= 1.

1, 2, 3, 4, 7, 8, 10, 15, 16, 31, 32, 36, 42, 63, 64, 127, 128, 136, 170, 255, 256, 292, 511, 512, 528, 682, 1023, 1024, 2047, 2048, 2080, 2184, 2340, 2730, 4095, 4096, 8191, 8192, 8256, 10922, 16383, 16384, 16912, 18724, 32767, 32768, 32896, 34952, 43690, 65535, 65536, 131071

Offset: 1

Ivan Neretin, May 10 2016

nonn

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:
()                  136: (4,4)
(1)                 170: (2,2,2,2)
(2)                 255: (1,1,1,1,1,1,1,1)
(1,1)               256: (9)
(3)                 292: (3,3,3)
(1,1,1)             511: (1,1,1,1,1,1,1,1,1)
(4)                 512: (10)
(2,2)               528: (5,5)
(1,1,1,1)           682: (2,2,2,2,2)
(5)                1023: (1,1,1,1,1,1,1,1,1,1)
(1,1,1,1,1)        1024: (11)
(6)                2047: (1,1,1,1,1,1,1,1,1,1,1)
(3,3)              2048: (12)
(2,2,2)            2080: (6,6)
(1,1,1,1,1,1)      2184: (4,4,4)
(7)                2340: (3,3,3,3)
(1,1,1,1,1,1,1)    2730: (2,2,2,2,2,2)
(8)                4095: (1,1,1,1,1,1,1,1,1,1,1,1)
(End)

Ivan Neretin, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A064896, A139708.
Cf. A137706 (smallest number indexing a new Haar graph).
Compositions in standard order are A066099.
Strict compositions are ranked by A233564.
Cf. A000120, A027750, A070939, A098504, A124767, A164894, A228351, A238279.

N:= 10^6: # to get all terms <= N
R:= select(`<=`,{seq(seq(2^(n-1)*(2^(n*m)-1)/(2^n-1), m = 1 .. ilog2(2*N)/n), n = 1..ilog2(2*N))},N):
sort(convert(R,list)); # Robert Israel, May 10 2016

Flatten@Table[d = Reverse@Divisors[n]; 2^(d - 1)*(2^n - 1)/(2^d - 1), {n, 17}]

From Gus Wiseman, Apr 04 2020: (Start)
A333381(a(n)) = A027750(n).
For n > 0, A124767(a(n)) = 1.
If n is a power of two, A333628(a(n)) = 0, otherwise = 1.
A333627(a(n)) is a power of 2.
(End)

A357000 Number of non-isomorphic cyclic Haar graphs on 2*n nodes.

Original entry on oeis.org

1, 2, 3, 5, 5, 12, 9, 22, 21, 44, 29, 157, 73, 244, 367, 649, 521, 2624, 1609, 7385, 8867, 19400, 16769, 92529, 67553, 216274, 277191, 815557, 662369, 4500266, 2311469

Offset: 1

Pontus von Brömssen, Sep 08 2022

The first value of n for which a(n) < A002729(n) - 1 is n = 8. This is because the first counterexample to the bicirculant analog to Ádám's conjecture occurs for n = 8. In the terminology of Hladnik, Marušič, and Pisanski, the smallest integer pair (i,j) such that i and j are Haar equivalent (i.e., the cyclic Haar graphs with indices i and j are isomorphic) but not cyclically equivalent (see A357005) is (141,147). See also A357001 and A357002.

Terms a(1)-a(29) were found by generating the cyclic Haar graphs with indices in A333764, and filtering out isomorphic graphs using Brendan McKay's software nauty.

Milan Hladnik, Dragan Marušič, and Tomaž Pisanski, Cyclic Haar graphs, Discrete Mathematics 244 (2002), 137-152.
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A002729, A008965, A049287, A091696, A137706, A291166, A333764, A339502, A357001, A357002, A357003, A357004, A357005, A357006.

a(n) is the number of terms k of A137706 in the interval 2^(n-1) <= k < 2^n.
a(n) is the number of fixed points k of A357004 in the interval 2^(n-1) <= k < 2^n.
a(n) <= A002729(n)-1 <= A091696(n) <= A008965(n).

a(30) from Eric W. Weisstein, Jun 27 2023

a(31) from Eric W. Weisstein, Jun 28 2023

A357004 Smallest k for which the cyclic Haar graphs with indices k and n are isomorphic.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 11, 9, 11, 11, 15, 16, 17, 17, 19, 17, 19, 19, 23, 17, 19, 19, 23, 19, 23, 23, 31, 32, 33, 34, 35, 36, 37, 37, 39, 34, 37, 42, 43, 37, 45, 43, 47, 33, 35, 37, 39, 37, 43, 45, 47, 35, 39, 43, 47, 39, 47, 47, 63, 64, 65, 65, 67, 65

Offset: 1

Pontus von Brömssen, Sep 08 2022

nonn

The fixed points are the terms of A137706.

The number of fixed points n in the interval 2^(m-1) <= n < 2^m equals A357000(m).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A137706 (fixed points; or ordered list of distinct terms), A163382, A272919 (terms that occur only once), A357000, A357005.

a(a(n)) = a(n).
a(n) = A357005(n) for n <= 146, but a(147) = 141 < 147 = A357005(147).
a(n) <= A357005(n) <= A163382(n).

A357006 Numbers k that are the smallest of all numbers that are cyclically equivalent to k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 19, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 47, 63, 64, 65, 67, 69, 71, 75, 79, 95, 127, 128, 129, 130, 131, 133, 135, 136, 137, 138, 139, 141, 143, 147, 151, 153, 155, 159, 170, 171, 175, 187, 191, 255, 256

Offset: 1

Pontus von Brömssen, Sep 08 2022

nonn

For the definition of cyclic equivalence, see A357005, or Hladnik, Marušič, and Pisanski (2002).
The sequence consists of the fixed points of A357005.
The number of terms k in the interval 2^(m-1) <= k < 2^m equals A002729(m)-1.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Milan Hladnik, Dragan Marušič, and Tomaž Pisanski, Cyclic Haar graphs, Discrete Mathematics 244 (2002), 137-152.

Cf. A002729, A137706 (subsequence), subsequence of A333764, A357005.

cp's OEIS Frontend

A272919 Numbers of the form 2^(n-1)(2^(nm)-1)/(2^n-1), n >= 1, m >= 1.

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A357000 Number of non-isomorphic cyclic Haar graphs on 2*n nodes.

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A357004 Smallest k for which the cyclic Haar graphs with indices k and n are isomorphic.

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A357006 Numbers k that are the smallest of all numbers that are cyclically equivalent to k.

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cp's OEIS Frontend

A272919 Numbers of the form 2^(n-1)*(2^(n*m)-1)/(2^n-1), n >= 1, m >= 1.

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A357000 Number of non-isomorphic cyclic Haar graphs on 2*n nodes.

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A357004 Smallest k for which the cyclic Haar graphs with indices k and n are isomorphic.

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A357006 Numbers k that are the smallest of all numbers that are cyclically equivalent to k.

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A272919 Numbers of the form 2^(n-1)(2^(nm)-1)/(2^n-1), n >= 1, m >= 1.