A004779 -id:A004779 - cp's OEIS frontend

A003796 Numbers with no 3 adjacent 0's in binary expansion.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.
Index entries for 2-automatic sequences.

Complement of A004779.
Cf. A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Cf. A063037, A007088.

a003796 n = a003796_list !! (n-1)
a003796_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 0 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,100],SequenceCount[IntegerDigits[#,2],{0,0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2015 *)

is(n)=while(n>7,if(bitand(n,7)==0,return(0));n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 9.829256652701616366441622119246549956902006567009112470631751387637507184399... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004781 Binary expansion contains 3 adjacent 1's.

Original entry on oeis.org

7, 14, 15, 23, 28, 29, 30, 31, 39, 46, 47, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 78, 79, 87, 92, 93, 94, 95, 103, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 142, 143, 151, 156, 157, 158, 159, 167

Offset: 1

N. J. A. Sloane

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

Complement of A003726.

Cf. A004779, A136037.

a004781 n = a004781_list !! (n - 1)
a004781_list = filter f [0..] where
   f x | x < 7     = False
       | otherwise = (x `mod` 8) == 7 || f (x `div` 2)
-- Reinhard Zumkeller, Jun 03 2012

q:= n-> verify([1$3], Bits[Split](n), 'sublist'):
select(q, [$0..200])[];  # Alois P. Heinz, Oct 22 2021

Select[Range[200],MemberQ[Partition[IntegerDigits[#,2],3,1], {1,1,1}]&]  (* Harvey P. Dale, Mar 31 2011 *)
Select[Range[200], StringContainsQ[IntegerString[#, 2], "111"] &] (* Amiram Eldar, Oct 22 2021 *)
Select[Range[200],SequenceCount[IntegerDigits[#,2],{1,1,1}]>0&] (* Harvey P. Dale, Dec 28 2021 *)

is(n)=!!bitand(bitand(n,n<<1),n<<2) \\ Charles R Greathouse IV, Sep 24 2012

a(n) ~ n. - Charles R Greathouse IV, Sep 24 2012

Offset corrected by Reinhard Zumkeller, Jun 03 2012

A136037 Numbers with at least three adjacent equal digits in binary representation.

Original entry on oeis.org

7, 8, 14, 15, 16, 17, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 46, 47, 48, 49, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 78, 79, 80, 81, 87, 88, 92, 93, 94, 95, 96, 97, 98, 99, 103, 104, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

Reinhard Zumkeller, Dec 11 2007

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A007088.
Complement of A063037; union of A004779 and A004781.
Supersequence of A037970.

Select[Range[150],SequenceCount[IntegerDigits[#,2],{x_,x_,x_}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2020 *)

def ok(n): b = bin(n)[2:]; return "000" in b or "111" in b
print(list(filter(ok, range(120)))) # Michael S. Branicky, Jul 08 2021

A300302 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation contains the binary representation of n as a substring.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Mar 08 2018

Each positive number k appears A122953(k) times in this array.

			Square array begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    2    3    4    5    6    7    8    9   10  <--  A000027
    2|    2    4    5    6    8    9   10   11   12   13  <--  A062289
    3|    3    6    7   11   12   13   14   15   19   22  <--  A004780
    4|    4    8    9   12   16   17   18   19   20   24  <--  A004753
    5|    5   10   11   13   20   21   22   23   26   27  <--  A004748
    6|    6   12   13   14   22   24   25   26   27   28  <--  A004749
    7|    7   14   15   23   28   29   30   31   39   46  <--  A004781
    8|    8   16   17   24   32   33   34   35   40   48  <--  A004779
    9|    9   18   19   25   36   37   38   39   41   50
   10|   10   20   21   26   40   41   42   43   52   53  <--  A132782

Rémy Sigrist, Perl program for A300302

Cf. A000027, A004748, A004749, A004753, A004779, A004780, A004781, A062289, A122953, A132782.

Perl
```
See Links section.
```

T(n, 1) = n.
T(n, 2) = 2*n.
T(n, 3) = 2*n + 1.
T(1, n) = A000027(n).
T(2, n) = A062289(n).
T(3, n) = A004780(n).
T(4, n) = A004753(n).
T(5, n) = A004748(n).
T(6, n) = A004749(n).
T(7, n) = A004781(n).
T(8, n) = A004779(n-1).
T(10, n) = A132782(n).

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A003796 Numbers with no 3 adjacent 0's in binary expansion.

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A004781 Binary expansion contains 3 adjacent 1's.

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A136037 Numbers with at least three adjacent equal digits in binary representation.

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A300302 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation contains the binary representation of n as a substring.

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