A076877 -id:A076877 - cp's OEIS frontend

A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Henry Bottomley, Mar 22 2000

Except for the initial term, 2^n appears 2^n times. - Lekraj Beedassy, May 26 2005
a(n) is the smallest k such that row k in triangle A265705 contains n. - Reinhard Zumkeller, Dec 17 2015
a(n) is the sum of totient function over powers of 2 <= n. - Anthony Browne, Jun 17 2016
Given positive n, reverse the bits of n and divide by 2^floor(log_2 n). Numerators are in A030101. Ignoring the initial 0, denominators are in this sequence. - Alonso del Arte, Feb 11 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

See A000035 for least significant bit(n).
MASKTRANS transform of A055975 (prepended with 0), MASKTRANSi transform of A048678.
Bisection of A065267, A065279, A065291, A072376.
First differences of A063915. Cf. A076877, A073121.
This is Guy Steele's sequence GS(5, 5) (see A135416).
Equals for n >= 1 the first right hand column of A160464. - Johannes W. Meijer, May 24 2009
Diagonal of A088370. - Alois P. Heinz, Oct 28 2011
Cf. A265705, A000010.

a053644 n = if n <= 1 then n else 2 * a053644 (div n 2)
-- Reinhard Zumkeller, Aug 28 2014
a053644_list = 0 : concat (iterate (\zs -> map (* 2) (zs ++ zs)) [1])
-- Reinhard Zumkeller, Dec 08 2012, Oct 21 2011, Oct 17 2010

[0] cat [2^Ilog2(n): n in [1..90]]; // Vincenzo Librandi, Dec 11 2018

a:= n-> 2^ilog2(n):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 20 2016

A053644[n_] := 2^(Length[ IntegerDigits[n, 2]] - 1); A053644[0] = 0; Table[A053644[n], {n, 0, 74}] (* Jean-François Alcover, Dec 01 2011 *)
nv[n_] := Module[{c = 2^n}, Table[c, {c}]]; Join[{0}, Flatten[Array[nv, 7, 0]]] (* Harvey P. Dale, Jul 17 2012 *)

a(n)=my(k=1);while(k<=n,k<<=1);k>>1 \\ Charles R Greathouse IV, May 27 2011

a(n) = if(!n, 0, 2^exponent(n)) \\ Iain Fox, Dec 10 2018

def a(n): return 0 if n==0 else 2**(len(bin(n)[2:]) - 1) # Indranil Ghosh, May 25 2017

def A053644(n): return 1<Chai Wah Wu, Jul 27 2022

(0 to 127).map(Integer.highestOneBit()) // _Alonso del Arte, Feb 26 2020

a(n) = a(floor(n / 2)) * 2.
a(n) = 2^A000523(n).
From n >= 1 onward, A053644(n) = A062383(n)/2.
a(0) = 0, a(1) = 1 and a(n+1) = a(n)*floor(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1 - x) * (x + Sum_{k >= 1} 2^(k - 1)*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = (A003817(n) + 1)/2 = A091940(n) + 1. - Reinhard Zumkeller, Feb 15 2004
a(n) = Sum_{k = 1..n} (floor(2^k/k) - floor((2^k - 1)/k))*A000010(k). - Anthony Browne, Jun 17 2016
a(2^m+k) = 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016

A062383 a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128

Offset: 0

Antti Karttunen, Jun 19 2001

Informally, write down 1 followed by 2^k 2^(k-1) times, for k = 1,2,3,4,... These are the denominators of the binary van der Corput sequence (see A030101 for the numerators). - N. J. A. Sloane, Dec 01 2019
a(n) is the denominator of the form 2^k needed to make the ratio (2n-1)/2^k lie in the interval [1-2], i.e. such ratios are 1/1, 3/2, 5/4, 7/4, 9/8, 11/8, 13/8, 15/8, 17/16, 19/16, 21/16, ... where the numerators are A005408 (The odd numbers).
Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the diagonal and 1 elsewhere. For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /. The order of this matrix as an element of GL(n,2) is a(n-1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001
A006257(n)/a(n) = (0, 0.1, 0.01, 0.11, 0.001, ...) enumerates all binary fractions in the unit interval [0, 1). - Fredrik Johansson, Aug 14 2006
a(n) = maximum of row n+1 in A240769. - Reinhard Zumkeller, Apr 13 2014
This is the discriminator sequence for the odious numbers. - N. J. A. Sloane, May 10 2016
From Jianing Song, Jul 05 2025: (Start)
a(n) is the period of {binomial(N,n) mod 2: N in Z}. For the general result, see A349593.
Since the modulus (2) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem. (End)

Harry J. Smith, Table of n, a(n) for n = 0..1000
L. K. Arnold, S. J. Benkoski and B. J. McCabe, The discriminator (a simple application of Bertrand's postulate). Amer. Math. Monthly 92 (1985), 275-277.
Sajed Haque, Chapter 2.6.1 of Discriminators of Integer Sequences, 2017, See p. 33.
S. Haque and J. Shallit, Discriminators and k-regular sequences, arXiv:1605.00092 [cs.DM], 2016.
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, Dec 08, 2020
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Wikipedia, Lucas's theorem
Index to divisibility sequences

Apart from the initial term, equals 2 * A053644. MASKTRANSi(A062383) seems to give a signed form of A038712. (See identities at A053644). floor_log_2 given in A054429.
Equals A003817(n)+1. Cf. A002884.
Bisection of A065285. Cf. A076877.
Equals for n>=1 the r(n) sequence of A160464. - Johannes W. Meijer, May 24 2009
Equals the r(n) sequence of A162440 for n>=1. - Johannes W. Meijer, Jul 06 2009
Cf. A030101, A264618, A264619.
Discriminator of the odious numbers (A000069). - Jeffrey Shallit, May 08 2016
Column 2 of A349593. A064235 (if offset 0), A385552, A385553, and A385554 are respectively columns 3, 5, 6, and 10.

import Data.List (transpose)
a062383 n = a062383_list !! n
a062383_list = 1 : zs where
   zs = 2 : (map (* 2) $ concat $ transpose [zs, zs])
-- Reinhard Zumkeller, Aug 27 2014, Mar 13 2014

[2^Floor(Log(2,2*n+1)): n in [0..70]]; // Bruno Berselli, Mar 04 2016

[seq(2^(floor_log_2(j)+1),j=0..127)]; or [seq(coerce1st_octave((2*j)+1),j=0..127)]; or [seq(a(j),j=0..127)];
coerce1st_octave := proc(r) option remember; if(r < 1) then coerce1st_octave(2*r); else if(r >= 2) then coerce1st_octave(r/2); else (r); fi; fi; end;
A062383 := proc(n)
    option remember;
    if n = 0 then
        1 ;
    else
        2*procname(floor(n/2));
    end if;
end proc:
A062383 := n -> 1 + Bits:-Iff(n, n):
seq(A062383(n), n=0..69); # Peter Luschny, Sep 23 2019

a[n_] := a[n] = 2 a[n/2 // Floor]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2016 *)
Table[2^Floor[Log2[n] + 1], {n, 0, 20}] (* Eric W. Weisstein, Nov 17 2017 *)
2^Floor[Log2[Range[0, 20]] + 1] (* Eric W. Weisstein, Nov 17 2017 *)
2^BitLength[Range[0, 100]] (* Paolo Xausa, Jan 29 2025 *)

{ a=1; for (n=0, 1000, write("b062383.txt", n, " ", a*=ceil((n + 1)/a)) ) } \\ Harry J. Smith, Aug 06 2009

a(n)=1<<(log(2*n+1)\log(2)) \\ Charles R Greathouse IV, Dec 08 2011

def A062383(n): return 1 << n.bit_length() # Chai Wah Wu, Jun 30 2022

a(1) = 1 and a(n+1) = a(n)*ceiling(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1-x) * (1 + Sum_{k>=0} 2^k*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = A142151(2*n)/2 + 1. - Reinhard Zumkeller, Jul 15 2008
log(a(n))/log(2) = A029837(n+1). - Johannes W. Meijer, Jul 06 2009
a(n+1) = a(n) + A099894(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = A264619(n) - A264618(n). - Reinhard Zumkeller, Dec 01 2015
a(n) is the smallest power of 2 > n. - Chai Wah Wu, Nov 04 2016
a(n) = 2^ceiling(log_2(n+1)). - M. F. Hasler, Sep 20 2017

A004754 Numbers n whose binary expansion starts 10.

Original entry on oeis.org

2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 64, 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, 93, 94, 95, 128, 129, 130, 131

Offset: 1

N. J. A. Sloane

A000120(a(n)) = A000120(n); A023416(a(n-1)) = A008687(n) for n > 1. - Reinhard Zumkeller, Dec 04 2015

			10 in binary is 1010, so 10 is in sequence.

T. D. Noe, Table of n, a(n) for n = 1..1023
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

Cf. A123001 (binary version), A004755 (11), A004756 (100), A004757 (101), A004758 (110), A004759 (111).
Cf. A004760, A053644, A062050, A076877.
Apart from initial terms, same as A004761.
Cf. A000120, A023416, A008687.

import Data.List (transpose)
a004754 n = a004754_list !! (n-1)
a004754_list = 2 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004754_list
-- Reinhard Zumkeller, Dec 04 2015

w = {1, 0}; Select[Range[2, 131], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* Michael De Vlieger, Aug 08 2016 *)

PARI
```
a(n)=n+2^floor(log(n)/log(2))
```

is(n)=n>1 && !binary(n)[2] \\ Charles R Greathouse IV, Sep 23 2012

def A004754(n): return n+(1<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + [n==0].
a(n) = n + 2^floor(log_2(n)) = n + A053644(n).
a(2^m+k) = 2^(m+1) + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A004755 Binary expansion starts 11.

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

N. J. A. Sloane

a(n) is the smallest value > a(n-1) (or > 1 for n=1) for which A001511(a(n)) = A001511(n). - Franklin T. Adams-Watters, Oct 23 2006

			12 in binary is 1100, so 12 is in the sequence.

Kenny Lau, Table of n, a(n) for n = 1..16383 (first 1023 terms from T. D. Noe)
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Equals union of A079946 and A080565.
Cf. A004754 (10), A004756 (100), A004757 (101), A004758 (110), A004759 (111).
Cf. A004760, A053644, A062050, A076877.

import Data.List (transpose)
a004755 n = a004755_list !! (n-1)
a004755_list = 3 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004755_list
-- Reinhard Zumkeller, Dec 04 2015

a:= proc(n) n+2*2^floor(log(n)/log(2)) end: seq(a(n),n=1..60); # Muniru A Asiru, Oct 16 2018

Flatten[Table[FromDigits[#,2]&/@(Join[{1,1},#]&/@Tuples[{0,1},n]),{n,0,5}]] (* Harvey P. Dale, Feb 05 2015 *)

PARI
```
a(n)=n+2*2^floor(log(n)/log(2))
```

is(n)=n>2 && binary(n)[2] \\ Charles R Greathouse IV, Sep 23 2012

f = open('b004755.txt', 'w')
lo = 3
hi = 4
i = 1
while i<16384:
    for x in range(lo,hi):
        f.write(str(i)+" "+str(x)+"\n")
        i += 1
    lo <<= 1
    hi <<= 1
# Kenny Lau, Jul 05 2016

def A004755(n): return n+(1<Chai Wah Wu, Jul 13 2022

a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1 + 2*[n==0].
a(n) = n + 2 * 2^floor(log_2(n)) = A004754(n) + A053644(n).
a(n) = 2n + A080079(n). - Benoit Cloitre, Feb 22 2003
G.f.: (1/(1+x)) * (1 + Sum_{k>=0, t=x^2^k} 2^k*(2t+t^2)/(1+t)).
a(n) = n + 2^(floor(log_2(n)) + 1) = n + A062383(n). - Franklin T. Adams-Watters, Oct 23 2006
a(2^m+k) = 2^(m+1) + 2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A004757 Binary expansion starts 101.

Original entry on oeis.org

5, 10, 11, 20, 21, 22, 23, 40, 41, 42, 43, 44, 45, 46, 47, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185

Offset: 1

N. J. A. Sloane

			22 in binary is 10110, so 22 is in sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..4095

Cf. A004754 (10), A004755 (11), A004756 (100), A004758 (110), A004759 (111).

Cf. A004760, A053644, A062050, A076877.

import Data.List (transpose)
a004757 n = a004757_list !! (n-1)
a004757_list = 5 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004757_list
-- Reinhard Zumkeller, Dec 04 2015

Table[n + 4*2^Floor@ Log2@ n, {n, 57}] (* or *)
w = {1, 0, 1}; Select[Range[5, 185], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* Michael De Vlieger, Aug 10 2016 *)
Select[Range[5,200],Take[IntegerDigits[#,2],3]=={1,0,1}&] (* Harvey P. Dale, Aug 26 2016 *)

PARI
```
a(n)=n+4*2^floor(log(n)/log(2))
```

def A004757(n): return n+(2<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 4*[n==0].
a(n) = n + 4 * 2^floor(log_2(n)) = A004756(n) + A053644(n).
a(2^m+k) = 5*2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A004758 Binary expansion starts 110.

Original entry on oeis.org

6, 12, 13, 24, 25, 26, 27, 48, 49, 50, 51, 52, 53, 54, 55, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213

Offset: 1

N. J. A. Sloane

			26 in binary is 11010, so 26 is in sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..4095

Cf. A004754 (10), A004755 (11), A004756 (100), A004757 (101), A004759 (111).

Cf. A004760, A053644, A062050, A076877.

import Data.List (transpose)
a004758 n = a004758_list !! (n-1)
a004758_list = 6 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004758_list
-- Reinhard Zumkeller, Dec 03 2015

w = {1, 1, 0}; Select[Range[5, 213], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* or *)
Table[n + 5*2^Floor@ Log2@ n, {n, 53}] (* Michael De Vlieger, Aug 10 2016 *)

PARI
```
a(n)=n+5*2^floor(log(n)/log(2))
```

def A004758(n): return n+(5<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 5*[n==0].
a(n) = n + 5 * 2^floor(log_2(n)) = A004757(n) + A053644(n).
a(2^m+k) = 6*2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A004756 Binary expansion starts 100.

Original entry on oeis.org

4, 8, 9, 16, 17, 18, 19, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153

Offset: 1

N. J. A. Sloane

			18 in binary is 10010, so 18 is in sequence.

T. D. Noe, Table of n, a(n) for n = 1..1023
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A004754 (10), A004755 (11), A004757 (101), A004758 (110), A004759 (111).

Cf. A004760, A053644, A062050, A076877.

import Data.List (transpose)
a004756 n = a004756_list !! (n-1)
a004756_list = 4 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004756_list
-- Reinhard Zumkeller, Dec 04 2015

Select[Range[4, 153], Take[IntegerDigits[#, 2], 3] == {1, 0, 0} &] (* Michael De Vlieger, Aug 07 2016 *)

PARI
```
a(n)=n+3*2^floor(log(n)/log(2))
```

def A004756(n): return n+(3<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 3*[n==0].
a(n) = n + 3 * 2^floor(log_2(n)) = A004755(n) + A053644(n).
a(2^m+k) = 2^(m+2) + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016

Edited by Ralf Stephan, Oct 12 2003

A004759 Binary expansion starts 111.

Original entry on oeis.org

7, 14, 15, 28, 29, 30, 31, 56, 57, 58, 59, 60, 61, 62, 63, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244

Offset: 1

N. J. A. Sloane

This is the minimal recursive sequence such that a(1)=7, A007814(a(n))= A007814(n) and A010060(a(n))=A010060(n). - Vladimir Shevelev, Apr 23 2009

			30 in binary is 11110, so 30 is in sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..4095

Cf. A004754 (10), A004755 (11), A004756 (100), A004757 (101), A004758 (110).
Cf. A004760, A053644, A062050, A076877.
Cf. A007814, A010060, A103204, A159559, A159560, A159615, A159619, A159629, A159698. -Vladimir Shevelev, Apr 23 2009

import Data.List (transpose)
a004759 n = a004759_list !! (n-1)
a004759_list = 7 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004759_list
-- Reinhard Zumkeller, Dec 03 2015

w = {1, 1, 1}; Select[Range[5, 244], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* Michael De Vlieger, Aug 10 2016 *)
Sort[FromDigits[#,2]&/@(Flatten[Table[Join[{1,1,1},#]&/@Tuples[{1,0},n],{n,0,5}],1])] (* Harvey P. Dale, Sep 01 2016 *)

PARI
```
a(n)=n+6*2^floor(log(n)/log(2))
```

def A004759(n): return n+(3<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 6[n==0].
a(n) = n + 6 * 2^floor(log_2(n)) = A004758(n) + A053644(n).
a(n+1) = min{m > a(n): A007814(m) = A007814(n+1) and A010060(m) = A010060(n+1)}. a(2^k) - a(2^k-1) = A103204(k+2), k >= 1. - Vladimir Shevelev, Apr 23 2009
a(2^m+k) = 7*2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

cp's OEIS Frontend

A053644 Most significant bit of n, msb(n); largest power of 2 less than or equal to n; write n in binary and change all but the first digit to zero.

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A062383 a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).

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A004754 Numbers n whose binary expansion starts 10.

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A004755 Binary expansion starts 11.

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A004757 Binary expansion starts 101.

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