A037014 -id:A037014 - cp's OEIS frontend

A037013 Reading binary expansion from right to left, run lengths strictly decrease.

0, 1, 3, 4, 7, 8, 15, 16, 24, 31, 32, 39, 48, 63, 64, 79, 96, 112, 127, 128, 143, 159, 192, 224, 255, 256, 287, 319, 384, 399, 448, 480, 511, 512, 543, 575, 624, 639, 768, 799, 896, 960, 1023, 1024, 1087, 1151, 1248, 1279, 1536, 1567, 1599, 1792, 1920, 1984, 2047, 2048, 2111

Offset: 1

N. J. A. Sloane

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

Subsequence of A037014, cf. A037015, A037016.

import Data.List (unfoldr, group)
a037013 n = a037013_list !! (n-1)
a037013_list = 0 : filter
   (all (< 0) . (\x -> zipWith (-) (tail $ rls x) $ rls x)) [1..] where
       rls = map length . group . unfoldr
             (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Mar 10 2012

Select[Range[0,2200],Min[Differences[Length/@Split[ IntegerDigits[ #,2]]]]>0&] (* Harvey P. Dale, Dec 17 2012 *)

More terms from Patrick De Geest, Feb 15 1999

Offset fixed and missing 1024 and 2047 inserted by Reinhard Zumkeller, Mar 10 2012

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

Original entry on oeis.org

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

N. J. A. Sloane

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

			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)

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

Subsequence of A037016, cf. A037013, A037014.

Cf. A030308.

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

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 *)

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

More terms from Patrick De Geest, Feb 15 1999

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

A037016 Numbers n with property that reading binary expansion from right to left (least significant to most significant), run lengths do not decrease.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 12, 13, 14, 15, 21, 25, 26, 28, 29, 30, 31, 42, 50, 51, 53, 56, 57, 58, 60, 61, 62, 63, 85, 101, 102, 106, 113, 114, 115, 117, 120, 121, 122, 124, 125, 126, 127, 170, 202, 204, 205, 213, 226, 227, 229, 230, 234, 240, 241, 242, 243, 245, 248

Offset: 1

N. J. A. Sloane

There are A000041(k) elements of this list consisting of k bits: a partition of k written in nonincreasing order corresponds to the binary expansion which when read left to right has run lengths as listed in the partition (reading left to right forces the initial run to be of 1s). - Jason Kimberley, Feb 08 2013
This sequence is a subsequence of A061854 (if we allow the initial 0 to be represented by the empty bit string). - Jason Kimberley, Feb 08 2013
The positive entries are those n for which row n of A101211 is weakly decreasing. Example: 6 is in the sequence because row 6 of A101211 is [2,1]; 8 is not in the sequence because row 8 of A101211 is [1,3]. - Emeric Deutsch, Jan 21 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A037015 (subsequence), A037014, A037013.

import Data.List (unfoldr, group)
a037016 n = a037016_list !! (n-1)
a037016_list = 0 : filter
   (all (>= 0) . (\x -> zipWith (-) (tail $ rls x) $ rls x)) [1..] where
       rls = map length . group . unfoldr
             (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Mar 10 2012

Select[ Range[0, 250], OrderedQ[ Reverse[ Length /@ Split[ IntegerDigits[#, 2] ] ] ]&] (* Jean-François Alcover, Apr 05 2013 *)

More terms from Patrick De Geest, Feb 15 1999

Offset fixed by Reinhard Zumkeller, Mar 10 2012

A335834 Sort the run lengths in binary expansion of n in ascending order (with multiplicities).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 7, 8, 11, 10, 11, 12, 11, 8, 15, 16, 23, 20, 19, 20, 21, 20, 23, 24, 19, 20, 19, 24, 23, 16, 31, 32, 47, 40, 39, 44, 43, 44, 39, 40, 43, 42, 43, 44, 43, 40, 47, 48, 39, 44, 51, 44, 43, 44, 39, 56, 39, 40, 39, 48, 47, 32, 63, 64, 95, 80, 79

Offset: 0

Rémy Sigrist, Jun 26 2020

This sequence preserves the number of runs (A005811) and the length (A070939) of the binary representation of a number.

			For n = 72:
- the binary representation of 72 is "1001000",
- the corresponding run lengths are: 1, 2, 1, 3,
- in ascending order: 1, 1, 2, 3,
- so the binary representation of a(72) is "1011000",
- and a(72) = 88.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A005811, A037014 (fixed points), A070939, A101211, A335835.

Array[FromDigits[Flatten@ MapIndexed[ConstantArray[Mod[First[#2], 2], #1] &, Sort[Length /@ Split[IntegerDigits[#, 2]]]], 2] &, 67] (* Michael De Vlieger, Jun 27 2020 *)

torl(n) = { my (rr=[]); while (n, my (r=valuation(n+(n%2), 2)); rr = concat(r, rr); n\=2^r); rr }
fromrl(rr) = { my (v=0); for (k=1, #rr, v = (v+(k%2))*2^rr[k]-(k%2)); v }
a(n) = { fromrl(vecsort(torl(n))) }

a(a(n)) = a(n).

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A037013 Reading binary expansion from right to left, run lengths strictly decrease.

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

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A037016 Numbers n with property that reading binary expansion from right to left (least significant to most significant), run lengths do not decrease.

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Haskell

Mathematica

Extensions

A335834 Sort the run lengths in binary expansion of n in ascending order (with multiplicities).

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