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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base
• nice

Comments

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

Examples

			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)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to binary expansion of n

Crossrefs

Subsequence of A037016, cf. A037013, A037014.

Cf. A030308.

Programs

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

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 *)

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

Extensions

More terms from Patrick De Geest, Feb 15 1999

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