A004756 -id:A004756 - 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

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

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

A304588 Length of shortest prefix of the Thue-Morse word (A010060) such that some length-n block appears twice.

Original entry on oeis.org

3, 5, 9, 10, 17, 18, 19, 20, 33, 34, 35, 36, 37, 38, 39, 40, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 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, 154, 155, 156, 157, 158, 159, 160, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266

Offset: 1

Jeffrey Shallit, May 15 2018

nonn

lim inf a(n)/n = 5/2 and lim sup a(n)/n = 4.

a(n) is "2-sychronized", which means that there is an automaton that accepts, in parallel, the base-2 expansions of n and a(n). For this sequence an 8-state automaton suffices. - Jeffrey Shallit, Mar 06 2020

			For n = 3 we have a(3) = 9 because the first 9 symbols of Thue-Morse are 011010011, and 011 is the first length-3 prefix to be repeated in this prefix.

Rémy Sigrist, Table of n, a(n) for n = 1..4096
Yann Bugeaud and Dong Han Kim, On the b-ary expansions of log(1+1/a) and e, Arxiv preprint arXiv:1510.00282 [math.NT], October 1 2015.
Yann Bugeaud and Dong Han Kim, On the b-ary expansions of log(1+1/a) and e, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 931-947.
Rémy Sigrist, PARI program for A304588

Cf. A010060, A004756.

PARI
```
See Links section.
```

Apparently, a(n+1) = A004756(n) + 1. - Rémy Sigrist, Nov 04 2020

A004762 Numbers whose binary expansion does not begin 100.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98

Offset: 1

N. J. A. Sloane

L. Sze, Sequence 102

Complementary to A004756.

Cf. A099396.

Join[{0,1,2,3},Select[Range[4,100],Take[IntegerDigits[#,2],3]!={1,0,0}&]] (* Harvey P. Dale, Feb 02 2015 *)

a(n+1) = n + 2^A099396(n).
G.f.: x/(1-x)^2 + x/(1-x) * Sum[k>=0, 2^k*x^(3*2^k)].
a(1) = 0, a(2) = 1, a(3) = 2, a(4) = 3, a(2^(m+1) + 2^m + k + 2) = 2^(m+2) + 2^m + k, m >= 0, 0 <= k < (2^(m+1) + 2^m). - Yosu Yurramendi, Aug 08 2016

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).

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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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A004758 Binary expansion starts 110.

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A004759 Binary expansion starts 111.

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A304588 Length of shortest prefix of the Thue-Morse word (A010060) such that some length-n block appears twice.

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