A004758 -id:A004758 - cp's OEIS frontend

A004754 Numbers n whose binary expansion starts 10.

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

A090996 Number of leading 1's in binary expansion of n.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Feb 29 2004

Mirror of triangle A065120. See example. - Omar E. Pol, Oct 17 2013

a(n) is also the least part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

			In binary : 14=1110 and there are 3 leading 1's, so a(14)=3.
From _Omar E. Pol_, Oct 17 2013: (Start)
Written as an irregular triangle with row lengths A011782 the sequence begins:
0;
1;
1,2;
1,1,2,3;
1,1,1,1,2,2,3,4;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
Right border gives A001477. Row sums give A000225.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..65535 (first 10001 terms from Vincenzo Librandi)
Index entries for sequences related to binary expansion of n

a(n) = A007814(1+A030101(n)).

Cf. A279209, A279210, A360189.

a := proc(n) if type(log[2](n+1), integer) then log[2](n+1) else a(floor((1/2)*n)) end if end proc: seq(a(n), n = 0 .. 200); # Emeric Deutsch, Jul 24 2017
# second Maple program:
b:= proc(n, t) `if`(n=0, t,
      b(iquo(n, 2, 'm'), m*(t+1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..127);  # Alois P. Heinz, Mar 06 2023

Join[{0},Table[Length@First@Split@IntegerDigits[n,2],{n,30}]] (* Birkas Gyorgy, Mar 09 2011 *) (* adapted by Vincenzo Librandi, Dec 23 2016 *)

a(n) = if(n==0, 0); b=binary(n+1); if(hammingweight(b) == 1, #b-1, a(n\2)) \\ David A. Corneth, Jul 24 2017

a(n) = if(n==0, 0); my(b = binary(n), r = #b); for(i=2, #b, if(!b[i], return(i-1))); r \\ David A. Corneth, Jul 24 2017

a(2^k-1)=k; a(A004754(k))=1; a(A004758(k))=2.
a(2^k-1)=k; for any other n, a(n) = a(floor(n/2)).
a(n) = f(n, 0) with f(n, x) = if n < 2 then n + x else f([n/2], (x+1)*(n mod 2)). - Reinhard Zumkeller, Feb 02 2007
Conjecture: a(n) = w(n+1)*(w(n+1)-w(n)+1) - w(2^(w(n+1)+1)-n-1) for n>0, where w(n) = floor(log_2(n)), that is, A000523(n). - Velin Yanev, Dec 21 2016
a(n) = A360189(n-1,floor(log_2(n))). - Alois P. Heinz, Mar 06 2023

Edited and corrected by Franklin T. Adams-Watters, Apr 08 2006

Sequence had accidentally been shifted left by one step, which was corrected and term a(0)=0 added by Antti Karttunen, Jan 01 2007

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

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

A179755 a(n) = A014486(A179754(n)).

Original entry on oeis.org

50, 216, 868, 3492, 13976, 56472, 225880, 897624, 3590576, 14471600, 57886120, 229868968, 919477136, 3706264464, 14825053488, 58832739632, 235330977088, 948740144448, 3794960504136, 15061020431688, 60244082031104

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, Python program for computing this sequence and the associated images.
A. Karttunen, Terms a(1)-a(768) drawn as binary strings, one pixel per bit.
A. Karttunen, Terms a(1)-a(700) drawn as binary strings, 3x3 pixels per bit.

a(n+1) = A004758(A179757(n)). Cf. also A122242, A122245.

A122872 Table by antidiagonals, T(n,k) is k-th number that starts with n in binary representation.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 5, 6, 4, 5, 8, 7, 8, 5, 6, 9, 12, 9, 10, 6, 7, 10, 13, 16, 11, 12, 7, 8, 11, 14, 17, 20, 13, 14, 8, 9, 16, 15, 18, 21, 24, 15, 16, 9, 10, 17, 24, 19, 22, 25, 28, 17, 18, 10, 11, 18, 25, 32, 23, 26, 29, 32, 19, 20, 11, 12, 19, 26, 33, 40, 27, 30, 33, 36, 21, 22, 12

Offset: 1

Franklin T. Adams-Watters, Oct 23 2006

In rows n through 2n-1, every integer >= n occurs exactly once.

			Top left corner is:
1 2 3 4 5
2 4 5 8 9
3 6 7 12 13
4 8 9 16 17
5 10 11 20 21

Rows: A000027, A004754, A004755, A004756, A004757, A004758, A004759. Algebraically, A053645 would be row zero, minus A080079 would be row minus one. See also A053644.

T(n,1) = n; T(n,2k) = 2T(n,k); T(n,2k+1) = 2T(n,k) + 1. T(n,k) = k + (n-1) * 2^floor(log_2(k)) = k + (n-1)*A053644(k).

A318934 Numbers whose binary expansion begins with exactly two 1's.

Original entry on oeis.org

3, 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, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223

Offset: 1

N. J. A. Sloane, Sep 13 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

This is the union of {3} and A004758.

Cf. A007088.

Join[{3},Select[Range[4,250],Take[IntegerDigits[#,2],3]=={1,1,0}&]] (* Harvey P. Dale, Oct 15 2022 *)

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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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A090996 Number of leading 1's in binary expansion of n.

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A004757 Binary expansion starts 101.

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A004756 Binary expansion starts 100.

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A004759 Binary expansion starts 111.

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