A272919 - OEIS

A272919

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

182

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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)

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}]

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.

Sequence in context: A332579 A357490 A333778 * A343603 A285506 A188190

Adjacent sequences: A272916 A272917 A272918 * A272920 A272921 A272922

KEYWORD

nonn

AUTHOR

Ivan Neretin, May 10 2016

STATUS

approved