A272919 Numbers of the form 2^(n-1)*(2^(n*m)-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

Author

Ivan Neretin, May 10 2016

Keywords

• nonn

Comments

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)

Links

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

Crossrefs

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.

Programs

Maple

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

Mathematica

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

Formula

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)