A037015 Numbers n with property that, reading binary expansion of n from right to left, run lengths strictly increase.

0, 1, 3, 6, 7, 14, 15, 28, 30, 31, 57, 60, 62, 63, 120, 121, 124, 126, 127, 241, 248, 249, 252, 254, 255, 483, 496, 497, 504, 505, 508, 510, 511, 966, 993, 995, 1008, 1009, 1016, 1017, 1020, 1022, 1023, 1987, 1990, 2016, 2017, 2019, 2032, 2033, 2040, 2041, 2044

Offset

1,3

Comments

There are A000009(k) elements of this list consisting of k bits. - Jason Kimberley, Jan 22 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Index entries for sequences related to binary expansion of n

Example

From Jason Kimberley, Jan 30 2013: (Start)
Interleaved lines:
binary expansions,
corresponding run lengths (distinct partitions);
1,
1;
11,
2;
110, 111,
2,1; 3;
1110, 1111,
3,1; 4;
11100, 11110, 11111,
3,2; 4,1; 5;
111001, 111100, 111110, 111111,
3,2,1; 4,2; 5,1; 6;
1111000, 1111001, 1111100, 1111110, 1111111,
4,3; 4,2,1; 5,2; 6,1; 7;
11110001, 11111000, 11111001, 11111100, 11111110, 11111111
4,3,1; 5,3; 5,2,1; 6,2; 7,1; 8;
111100011, 111110000, 111110001, 111111000, 111111001, 111111100, 111111110, 111111111,
4,3,2; 5,4; 5,3,1; 6,3; 6,2,1; 7,2; 8,1; 9;
Notice the reversed sorting when a part corresponds to a run of 0's.
(End)

Mathematica

Select[Range[0, 2500], Min[Differences[Length/@Split[ Reverse[ IntegerDigits[ #, 2]]]]]>0&] (* Harvey P. Dale, Nov 18 2014 *)
Select[Range[0, 2100], Max[Differences[Length/@Split[IntegerDigits[#, 2]]]]<0&] (* Harvey P. Dale, Jun 28 2020 *)

Prog

(Haskell)
import Data.List (group)
a037015 n = a037015_list !! (n-1)
a037015_list = filter (all (> 0) . ds) [0..] where
   ds x = zipWith (-) (tail gs) gs where
      gs = map length $ group $ a030308_row x
-- Reinhard Zumkeller, Jul 31 2013, Mar 10 2012
(Python)
from itertools import groupby
A037015_list = []
for n in range(10**5):
    c = None
    for x, y in groupby(bin(n)[2:]):
        z = len(list(y))
        if c != None and z >= c:
            break
        c = z
    else:
        A037015_list.append(n) # Chai Wah Wu, Sep 14 2021

Crossrefs

Subsequence of A037016, cf. A037013, A037014.

Cf. A030308.

Sequence in context: A099403 A324726 A333379 * A138218 A056055 A359476

Adjacent sequences: A037012 A037013 A037014 * A037016 A037017 A037018

Keyword

nonn,easy,base,nice

Author

N. J. A. Sloane

Extensions

More terms from Patrick De Geest, Feb 15 1999

Offset fixed and missing 1023 inserted by Reinhard Zumkeller, Mar 10 2012

Status

approved