A062289 -id:A062289 - cp's OEIS frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

0, 3, 6, 5, 4, 15, 10, 9, 8, 11, 14, 13, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 126, 93, 76, 71, 66, 65, 64, 67, 70, 69

Offset: 0

Philippe Deléham, Feb 13 2005

All terms are distinct, but certain terms (see A102371) are missing. But see A103122.

Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.

			........0
........1
.......10
.......11
......100
......101
......110
......111
.....1000
.........
The upward-sloping diagonals are:
0
11
110
101
100
1111
1010
.......
giving 0, 3, 6, 5, 4, 15, 10, ...
The sequence has a natural decomposition into blocks (see the paper): 0; 3; 6, 5, 4; 15, 10, 9, 8, 11, 14, 13; 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30; 61, ...
Reading the array of binary numbers along diagonals with slope 1 gives this sequence, slope 2 gives A105085, slope 0 gives A001477 and slope -1 gives A105033.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
Index entries for sequences related to binary expansion of n

Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.
Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).
Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.
Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.
Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.

a102370 n = a102370_list !! n
a102370_list = 0 : map (a105027 . toInteger) a062289_list
-- Reinhard Zumkeller, Jul 21 2012

A102370:=proc(n) local t1,l; t1:=n; for l from 1 to n do if n+l mod 2^l = 0 then t1:=t1+2^l; fi; od: t1; end;

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 71}] (* Robert G. Wilson v, Mar 21 2005 *)

A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) \\ Benoit Cloitre, Mar 20 2005

{a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* Michael Somos, Mar 26 2012 */

def a(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]) # Indranil Ghosh, May 03 2017

a(n) = n + Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k. (Cf. A103185.) In particular, a(n) >= n. - N. J. A. Sloane, Mar 18 2005

a(n) = A105027(A062289(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005

A333766 Maximum part of the n-th composition in standard order. a(0) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 2, 2, 1, 5, 4, 3, 3, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 6, 5, 4, 4, 3, 3, 3, 3, 4, 3, 2, 2, 3, 2, 2, 2, 5, 4, 3, 3, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 7, 6, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 5, 4, 3, 3, 3, 2, 2

Offset: 0

Gus Wiseman, Apr 05 2020

nonn

One plus the longest run of 0's in the binary expansion of n.

A composition of n is a finite sequence of positive integers summing to n. 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.

			The 100th composition in standard order is (1,3,3), so a(100) = 3.

Positions of ones are A000225.
Positions of terms <= 2 are A003754.
The version for prime indices is A061395.
Positions of terms > 1 are A062289.
Positions of first appearances are A131577.
The minimum part is given by A333768.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Max@@stc[n]],{n,0,100}]

For n > 0, a(n) = A087117(n) + 1.

A038554 Derivative of n: write n in binary, replace each pair of adjacent bits with their mod 2 sum (a(0)=a(1)=0 by convention). Also n XOR (n shift 1).

Original entry on oeis.org

0, 0, 1, 0, 2, 3, 1, 0, 4, 5, 7, 6, 2, 3, 1, 0, 8, 9, 11, 10, 14, 15, 13, 12, 4, 5, 7, 6, 2, 3, 1, 0, 16, 17, 19, 18, 22, 23, 21, 20, 28, 29, 31, 30, 26, 27, 25, 24, 8, 9, 11, 10, 14, 15, 13, 12, 4, 5, 7, 6, 2, 3, 1, 0, 32, 33, 35, 34, 38, 39, 37, 36, 44, 45, 47, 46, 42, 43, 41, 40, 56, 57

Offset: 0

N. J. A. Sloane

From Antti Karttunen: this is also a version of A003188: a(n) = A003188(n) - 2^floor(log_2(A003188(n))), that is, the corresponding Gray code expansion, but with highest 1-bit turned off. Also a(n) = A003188(n) - 2^floor(log_2(n)).
From John W. Layman: {a(n)} is a self-similar sequence under Kimberling's "upper-trimming" operation.
a(A000225(n)) = 0; a(A062289(n)) > 0; a(A038558(n)) = n. - Reinhard Zumkeller, Mar 06 2013

			If n = 18 = 10010_2, derivative is (1+0)(0+0)(0+1)(1+0) = 1011_2, so a(18)=11.

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

T. D. Noe, Table of n, a(n) for n = 0..4096
Clark Kimberling, Fractal sequences.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
J. W. Layman, View the fractal-like graph of a(n) vs. n.
Ralf Stephan, Some divide-and-conquer sequences ....
Ralf Stephan, Table of generating functions.

Cf. A038570, A038571. See A003415 for another definition of the derivative of a number.
Cf. A038556 (rotates n instead of shifting).
Cf. A000035.
Cf. A030308.

import Data.Bits (xor)
a038554 n = foldr (\d v -> v * 2 + d) 0 $ zipWith xor bs $ tail bs
   where bs = a030308_row n
-- Reinhard Zumkeller, May 26 2013, Mar 06 2013

A038554 := proc(n) local i,b,ans; ans := 0; b := convert(n,base,2); for i to nops(b)-1 do ans := ans+((b[ i ]+b[ i+1 ]) mod 2)*2^(i-1); od; RETURN(ans); end; [ seq(A038554(i),i=0..100) ];

a[0] = a[1] = 0; a[n_ /; Mod[n, 4] == 0] := a[n] = 2*a[n/2]; a[n_ /; Mod[n, 4] == 1] := a[n] =  2*a[(n-1)/2] + 1; a[n_ /; Mod[n, 4] == 2] := a[n] = 2*a[n/2] + 1; a[n_ /; Mod[n, 4] == 3] := a[n] = 2*a[(n-1)/2]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Jul 13 2012, after Ralf Stephan *)
Table[FromDigits[Mod[Total[#],2]&/@Partition[IntegerDigits[n,2],2,1],2],{n,0,90}] (* Harvey P. Dale, Oct 27 2015 *)

a003188(n)=bitxor(n, n>>1);
a(n)=if(n<2, 0, a003188(n) - 2^logint(a003188(n), 2)); \\ Indranil Ghosh, Apr 26 2017

import math
def a003188(n): return n^(n>>1)
def a(n): return 0 if n<2 else a003188(n) - 2**int(math.floor(math.log(a003188(n), 2))) # Indranil Ghosh, Apr 26 2017

If 2*2^k <= n < 3*2^k then a(n) = 2^k + a(2^(k+2)-n-1); if 3*2^k <= n < 4*2^k then a(n) = a(n-2^(k+1)). - Henry Bottomley, May 11 2000
G.f.: (1/(1-x)) * Sum_{k>=0} 2^k*(t^4-t^3+t^2)/(1+t^2), t=x^2^k. - Ralf Stephan, Sep 10 2003
a(0)=0, a(2n) = 2*a(n) + [n odd], a(2n+1) = 2*a(n) + [n>0 even]. - Ralf Stephan, Oct 20 2003
a(0) = a(1) = 0, a(4n) = 2*a(2n), a(4n+2) = 2*a(2n+1)+1, a(4n+1) = 2*a(2n)+1, a(4n+3) = 2*a(2n+1). Proof by Nikolaus Meyberg following a conjecture by Ralf Stephan.

More terms from Erich Friedman

A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 2, 1, 1, 0, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 5, 4, 3, 3, 2, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 6, 5, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 4, 3, 3, 2, 2

Offset: 0

Reinhard Zumkeller, Aug 14 2003

The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k > 0; A087116(n) = 0; A023416(n) = 0.

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. Then a(k) is the maximum part of this composition, minus one. The maximum part is A333766(k). - Gus Wiseman, Apr 09 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A023416, A007088, A007814.
Cf. A025480, A038374, A090046, A090047, A090048, A090049, A090050.
Positions of zeros are A000225.
Positions of terms <= 1 are A003754.
Positions of terms > 0 are A062289.
Positions of first appearances are A131577.
The version for prime indices is A252735.
The proper maximum is A333766.
The version for minimum is A333767.
Maximum prime index is A061395.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A061395, A087117, A228351, A328594, A333217, A333218, A333219, A333767, A333768.

import Data.List (unfoldr, group)
a087117 0       = 1
a087117 n
  | null $ zs n = 0
  | otherwise   = maximum $ map length $ zs n where
  zs = filter ((== 0) . head) . group .
       unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, May 01 2012

A087117 := proc(n)
    local d,l,zlen ;
    if n = 0 then
        return 1 ;
    end if;
    d := convert(n,base,2) ;
    for l from nops(d)-1 to 0 by -1 do
        zlen := [seq(0,i=1..l)] ;
        if verify(zlen,d,'sublist') then
            return l ;
        end if;
    end do:
    return 0 ;
end proc; # R. J. Mathar, Nov 05 2012

nz[n_]:=Max[Length/@Select[Split[IntegerDigits[n,2]],MemberQ[#,0]&]]; Array[nz,110,0]/.-\[Infinity]->0 (* Harvey P. Dale, Sep 05 2017 *)

h(n)=if(n<2, return(0)); my(k=valuation(n,2)); if(k, max(h(n>>k), k), n++; n>>=valuation(n,2); h(n-1))
a(n)=if(n,h(n),1) \\ Charles R Greathouse IV, Apr 06 2022

a(n) = max(A007814(n), a(A025480(n-1))) for n >= 2. - Robert Israel, Feb 19 2017
a(2n+1) = a(n) (n>=1); indeed, the binary form of 2n+1 consists of the binary form of n with an additional 1 at the end - Emeric Deutsch, Aug 18 2017
For n > 0, a(n) = A333766(n) - 1. - Gus Wiseman, Apr 09 2020

A261461 a(n) is the smallest nonzero number that is not a substring of n in its binary representation.

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 4, 5, 4, 4, 2, 3, 3, 3, 5, 3, 3, 4, 4, 5, 5, 4, 4, 5, 4, 4, 2, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 4, 7, 4, 4, 4, 5, 5, 5, 5, 7, 4, 4, 4, 5, 5, 4, 4, 5, 4, 4, 2, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 6, 3, 3

Offset: 0

Reinhard Zumkeller, Aug 30 2015

A261018(n) = a(A260273(n)).
Is a(n) = A091460(n) for n>1? - R. J. Mathar, Sep 02 2015. The lowest counterexample occurs at a(121) = 5 < 6 = A091460(121). - Álvar Ibeas, Sep 08 2020
a(A062289(n))=A261922(A062289(n)); a(A126646(n))!=A261922(A126646(n)). - Reinhard Zumkeller, Sep 17 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program

Cf. A007088, A030308, A260273, A261018; record values and where they occur: A261466, A261467.
See A261922 for a variant.
Cf. A062289, A144016, A261922, A126646.

import Data.List (isInfixOf)
a261461 x = f $ tail a030308_tabf where
   f (cs:css) = if isInfixOf cs (a030308_row x)
                   then f css else foldr (\d v -> 2 * v + d) 0 cs

fQ[m_, n_] := Block[{g}, g[x_] := ToString@ FromDigits@ IntegerDigits[x, 2]; StringContainsQ[g@ n, g@ m]]; Table[k = 1; While[fQ[k, n] && k < n, k++]; k, {n, 85}] (* Michael De Vlieger, Sep 21 2015 *)

from itertools import count
def a(n):
    b, k = bin(n)[2:], 1
    return next(k for k in count(1) if bin(k)[2:] not in b)
print([a(n) for n in range(86)]) # Michael S. Branicky, Feb 26 2023

a(n) = A144016(n) + 1 for any n > 0. - Rémy Sigrist, Mar 10 2018

A101082 Numbers n such that binary representation contains bit strings "10" and "01" (possibly overlapping).

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Rick L. Shepherd, Nov 29 2004

Subsequence of A062289; set difference A062289 minus A043569.
Complement of A023758. Also numbers not the sum of consecutive powers of 2. - Omar E. Pol, Mar 04 2013
Equivalently, numbers not the difference of two powers of two. - Charles R Greathouse IV, Mar 07 2013
The terms >=9 are bases in which a power of 2 exists, which does not contain a digit that is a power of 2. In base 10, 2^16 = 65536 is such a number, as it does not contain any one-digit power of 2, which in base 10 are 1, 2, 4 and 8. - Patrick Wienhöft, Jul 28 2016

			29 = 11101_2 is a term, "10" and "01" are contained (here overlapping).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Patrick Wienhöft, Python program
Index entries for 2-automatic sequences.

Complement: A023758.

Cf. A043569, A062289, A049502.

a101082 n = a101082_list !! (n-1)
a101082_list = filter ((> 0) . a049502) [0..]
-- Reinhard Zumkeller, Jun 17 2015

Select[Range@ 120, Function[d, Times @@ Total@ Map[Map[Function[k, Boole@ MatchQ[#, k]], {{1, 0}, {0, 1}}] &, Partition[d, 2, 1]] > 0]@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Dec 23 2016 *)
Select[Range[100],With[{c=IntegerDigits[#,2]},SequenceCount[c,{1,0}]>0&&SequenceCount[c,{0,1}]>0]&] (* Harvey P. Dale, Jun 08 2025 *)

is(n)=n>>=valuation(n,2);n+1!=1<Charles R Greathouse IV, Mar 07 2013

def A101082(n):
    def f(x): return n+((k:=x.bit_length())*(k-1)>>1)+sum(1 for i in range(k) if (1<Chai Wah Wu, Feb 23 2025

a(n) ~ n. In particular a(n) = n + (log_2 n)^2/2 + O(log n). - Charles R Greathouse IV, Mar 07 2013

A049502(a(n)) > 0. - Reinhard Zumkeller, Jun 17 2015

A043545 (Maximal base-2 digit of n) - (minimal base-2 digit of n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Clark Kimberling

Characteristic function of A062289 (non-Mersenne numbers A000225). - Omar E. Pol, Sep 05 2021

			G.f. = x^2 + x^4 + x^5 + x^6 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Cf. A000225, A036987, A062289.

Column k=0 of A347519.

a043545 = (1 -) . a036987  -- Reinhard Zumkeller, Nov 02 2013

mb2d[n_]:=Module[{n2=IntegerDigits[n,2]},Max[n2]-Min[n2]]; Array[mb2d,120,0] (* Harvey P. Dale, Feb 24 2015 *)

{a(n) = if( n<0, 0, n++; n != 2^valuation(n, 2))}; /* Michael Somos, Aug 25 2003 */

a(n) = !!bitand(n, n+1); \\ Ruud H.G. van Tol, Sep 12 2023

0 followed by a string of 2^k - 1 1's. Also a(n)=0 iff n = 2^m - 1.
G.f.: 1/(1-x) - Sum_{k>=0} x^(2^k-1). - Michael Somos, Aug 25 2003
a(n) = 1 - A036987(n). 1's complement of Fredhold-Rueppel sequence. - Michael Somos, Aug 25 2003
a(n) = (1 + (-1)^binomial(n, floor(n/2)))/2. - Paul Barry, Jun 07 2006
Ignoring first zero and beginning instead with offset 2, a(n) = A006530(n) mod 2. - Rick L. Shepherd, Jun 09 2008
a(n) = A000777(n) mod 2, for n > 0. - John M. Campbell, Jul 16 2016

A105027 Write numbers in binary under each other; to get the next block of 2^k (k >= 0) terms of the sequence, start at 2^k, read diagonals in upward direction and convert to decimal.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 4, 7, 15, 10, 9, 8, 11, 14, 13, 12, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 29, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 62, 126, 93, 76, 71, 66, 65, 64, 67, 70

Offset: 0

N. J. A. Sloane, Apr 03 2005

This is a permutation of the nonnegative integers.
Structure: blocks of size 2^k - 1 taken from A102370, interspersed with terms of A102371. - Philippe Deléham, Nov 17 2007
a(A062289(n)) = A102370(n) for n > 0; a(A000225(n)) = A102371(n); a(A214433(n)) = A105025(a(n)). - Reinhard Zumkeller, Jul 21 2012

			        0
        1
       10
       11
   -> 100  Starting here, the upward diagonals
      101  read 110, 101, 100, 111, giving the block 6, 5, 4, 7.
      110
      111
     1000
     1001
     1010
     1011
      ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

Cf. A102370, A105025, A105026, A105028.
Cf. A070939, A000079.
Cf. A214414 (fixed points), A214417 (inverse).

import Data.Bits ((.|.), (.&.))
a105027 n = foldl (.|.) 0 $ zipWith (.&.)
                  a000079_list $ enumFromTo (n + 1 - a070939 n) n
-- Reinhard Zumkeller, Jul 21 2012

block[k_] := Module[{t}, t = Table[PadLeft[IntegerDigits[n, 2], k+1], {n, 2^(k-1), 2^(k+1)-1}]; Table[FromDigits[Table[t[[n-m+1, m]], {m, 1, k+1}], 2], {n,2^(k-1)+1, 2^(k-1)+2^k}]]; block[0] = {0, 1}; Table[block[k], {k, 0, 6}] // Flatten (* Jean-François Alcover, Jun 30 2015 *)

apply( {A105027(n,L=exponent(n+!n))=sum(k=0,L,bitand(n+k-L,2^k))}, [0..55]) \\ M. F. Hasler, Apr 18 2022

a(2^n - 1) = A102371(n) for n > 0. - Philippe Deléham, May 10 2005

More terms from John W. Layman, Apr 07 2005

A043569 Numbers whose base-2 representation has exactly 2 runs.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 16, 24, 28, 30, 32, 48, 56, 60, 62, 64, 96, 112, 120, 124, 126, 128, 192, 224, 240, 248, 252, 254, 256, 384, 448, 480, 496, 504, 508, 510, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1024, 1536, 1792, 1920, 1984, 2016, 2032, 2040, 2044

Offset: 1

Clark Kimberling

Numbers whose binary representation contains the bit string "10" but not "01". Subsequence of A062289; set difference A062289 minus A101082. - Rick L. Shepherd, Nov 29 2004

Mersenne numbers (A000225) times powers of 2 (A000079). Therefore this sequence contains the even perfect numbers (A000396). - Alonso del Arte, Apr 21 2006

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A004736, A007814, A062289, A065442, A101082.

a:=proc(n) local nn,nd: nn:=convert(n,base,2): nd:={seq(nn[j]-nn[j-1],j=2..nops(nn))}: if n=2 then 2 elif nd={0,1} then n else fi end: seq(a(n),n=1..2100); # Emeric Deutsch, Apr 21 2006

Take[Sort[Flatten[Table[(2^x - 1)*(2^y), {x, 32}, {y, 32}]]], 54] (* Alonso del Arte, Apr 21 2006 *)
Select[Range[2500],Length[Split[IntegerDigits[#,2]]]==2&] (* or *) Select[Range[2500],SequenceCount[IntegerDigits[#,2],{1,0}]>0 && SequenceCount[ IntegerDigits[#,2],{0,1}]==0&] (* Harvey P. Dale, Oct 04 2024 *)

def ok(n): b = bin(n)[2:]; return "10" in b and "01" not in b
print([m for m in range(2045) if ok(m)]) # Michael S. Branicky, Feb 04 2021

def a_next(a_n): t = a_n >> 1; return (a_n | t) + (t & 1)
a_n = 2; a = []
for i in range(54): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022

This sequence is twice A023758. - Franklin T. Adams-Watters, Apr 21 2006
Sum_{n>=1} 1/a(n) = A065442. - Amiram Eldar, Feb 20 2022
A007814(a(n)) = A004736(n). - Lorenzo Sauras Altuzarra, Feb 01 2023

More terms from Rick L. Shepherd, Nov 29 2004

A261922 a(n) = smallest nonnegative number that is not a substring of n in its binary representation.

Original entry on oeis.org

1, 0, 3, 0, 3, 3, 4, 0, 3, 3, 3, 4, 5, 4, 4, 0, 3, 3, 3, 5, 3, 3, 4, 4, 5, 5, 4, 4, 5, 4, 4, 0, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 4, 7, 4, 4, 4, 5, 5, 5, 5, 7, 4, 4, 4, 5, 5, 4, 4, 5, 4, 4, 0, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 6, 3, 3, 4, 4, 7, 7, 4, 4, 8, 4, 4, 4, 5, 5, 5, 5, 5, 7

Offset: 0

N. J. A. Sloane, Sep 16 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Similar to A261461.

Cf. A007088, A030308, A062289, A126646, A262281.

import Data.List (isInfixOf)
a261922 x = f a030308_tabf where
   f (cs:css) = if isInfixOf cs (a030308_row x)
                   then f css else foldr (\d v -> 2 * v + d) 0 cs
-- Reinhard Zumkeller, Sep 17 2015

bstr(n) = if (n==0, "0", my(s="", b=binary(n)); for (i=1, #b, s=concat(s, b[i])); s);
a(n) = my(sn=btostr(n), k=0); while (#strsplit(sn, bstr(k)) != 1, k++); k; \\ Michel Marcus, Sep 20 2023

def a(n): b=bin(n)[2:]; return next(k for k in range(2**len(b)) if bin(k)[2:] not in b)
print([a(n) for n in range(99)]) # Michael S. Branicky, Sep 21 2023

From Reinhard Zumkeller, Sep 17 2015: (Start)
a(A062289(n)) = A261461(A062289(n)).
a(A126646(n)) != A261461(A126646(n)). (End)

cp's OEIS Frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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A333766 Maximum part of the n-th composition in standard order. a(0) = 0.

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A038554 Derivative of n: write n in binary, replace each pair of adjacent bits with their mod 2 sum (a(0)=a(1)=0 by convention). Also n XOR (n shift 1).

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A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

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A261461 a(n) is the smallest nonzero number that is not a substring of n in its binary representation.

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A101082 Numbers n such that binary representation contains bit strings "10" and "01" (possibly overlapping).

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