A004753 -id:A004753 - cp's OEIS frontend

A003754 Numbers with no adjacent 0's in binary expansion.

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181

Offset: 1

N. J. A. Sloane

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]
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. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. See the corresponding example below. - Gus Wiseman, Apr 04 2020
The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - Peter Munn, Sep 06 2022

			21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
From _Gus Wiseman_, Apr 04 2020: (Start)
The terms together with the corresponding compositions begin:
    0: ()            30: (1,1,1,2)         90: (2,1,2,2)
    1: (1)           31: (1,1,1,1,1)       91: (2,1,2,1,1)
    2: (2)           42: (2,2,2)           93: (2,1,1,2,1)
    3: (1,1)         43: (2,2,1,1)         94: (2,1,1,1,2)
    5: (2,1)         45: (2,1,2,1)         95: (2,1,1,1,1,1)
    6: (1,2)         46: (2,1,1,2)        106: (1,2,2,2)
    7: (1,1,1)       47: (2,1,1,1,1)      107: (1,2,2,1,1)
   10: (2,2)         53: (1,2,2,1)        109: (1,2,1,2,1)
   11: (2,1,1)       54: (1,2,1,2)        110: (1,2,1,1,2)
   13: (1,2,1)       55: (1,2,1,1,1)      111: (1,2,1,1,1,1)
   14: (1,1,2)       58: (1,1,2,2)        117: (1,1,2,2,1)
   15: (1,1,1,1)     59: (1,1,2,1,1)      118: (1,1,2,1,2)
   21: (2,2,1)       61: (1,1,1,2,1)      119: (1,1,2,1,1,1)
   22: (2,1,2)       62: (1,1,1,1,2)      122: (1,1,1,2,2)
   23: (2,1,1,1)     63: (1,1,1,1,1,1)    123: (1,1,1,2,1,1)
   26: (1,2,2)       85: (2,2,2,1)        125: (1,1,1,1,2,1)
   27: (1,2,1,1)     86: (2,2,1,2)        126: (1,1,1,1,1,2)
   29: (1,1,2,1)     87: (2,2,1,1,1)      127: (1,1,1,1,1,1,1)
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
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.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
Wikipedia, Ahnentafel.
Witzel, Stefan On panel-regular ~A2 lattices Geom. Dedicata 191, 85-135 (2017).
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1) - a(n); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005
Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Complement of A004753.
Cf. A023705, A196168, A280873.
Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
Cf. A318928.

a003754 n = a003754_list !! (n-1)
a003754_list = filter f [0..] where
   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

isA003754 := proc(n) local bdgs ; bdgs := convert(n,base,2) ; for i from 2 to nops(bdgs) do if op(i,bdgs)=0 and op(i-1,bdgs)= 0 then return false; end if; end do; return true; end proc:
A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:
# R. J. Mathar, Oct 23 2010

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {_, 0, 0, _}]&] (* Jean-François Alcover, Oct 25 2011 *)
Select[Range[0,200],SequenceCount[IntegerDigits[#,2],{0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

i=0
while i<=500:
    if "00" not in bin(i)[2:]:
        print(str(i), end=',')
    i+=1 # Indranil Ghosh, Feb 11 2017

Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020

A328212 Lazy-Fibonacci-Niven numbers: numbers divisible by the number of terms in their lazy Fibonacci representation (A112310).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 16, 28, 30, 35, 36, 48, 55, 60, 70, 72, 75, 78, 84, 90, 102, 105, 114, 119, 126, 133, 144, 147, 154, 156, 161, 168, 180, 182, 184, 192, 198, 203, 208, 216, 224, 238, 240, 245, 252, 259, 264, 266, 272, 280, 296, 301, 304, 308, 315, 320, 322

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			6 is in the sequence since A112310(6) = 3 and 3 is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A004753, A005349, A112310.

ooQ[n_] := Module[{k = n}, While[k > 3, If[Divisible[k, 4], Return[True], k = Quotient[k, 2]]]; False]; c = 0; s = {}; Do[If[! ooQ[k], c++; d = Total @ IntegerDigits[k, 2]; If[Divisible[c, d], AppendTo[s, c]]], {k, 1, 2000}]; s

A004748 Binary expansion contains 101.

Original entry on oeis.org

5, 10, 11, 13, 20, 21, 22, 23, 26, 27, 29, 37, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 58, 59, 61, 69, 74, 75, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 101, 104, 105, 106, 107, 108, 109, 110, 111, 116, 117, 118, 119, 122

Offset: 1

N. J. A. Sloane

			20 is in the sequence because 20 = 10100_2. '10100' has '101' as one of its substrings. - _Indranil Ghosh_, Feb 11 2017

Indranil Ghosh, Table of n, a(n) for n = 1..50000
Index entries for 2-automatic sequences.

Cf. A004749, A004750, A004751, A004752, A004753.

Select[Range[200], MemberQ[Partition[IntegerDigits[#, 2], 3, 1], {1, 0, 1}] &] (* Vincenzo Librandi, Feb 17 2018 *)
Select[Range[200],SequenceCount[IntegerDigits[#,2],{1,0,1}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2020 *)

is(n)=while(n>4,if(bitand(n,7)==5, return(1)); n>>=1); 0 \\ Charles R Greathouse IV, Feb 11 2017

i=j=0
while j<=100:
    if "101" in bin(i)[2:]:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 11 2017

a(n) ~ n. - Charles R Greathouse IV, Oct 23 2015

Offset corrected by Charles R Greathouse IV, Feb 11 2017

A004749 Numbers whose binary expansion contains the substring '110'.

Original entry on oeis.org

6, 12, 13, 14, 22, 24, 25, 26, 27, 28, 29, 30, 38, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 70, 76, 77, 78, 86, 88, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113

Offset: 1

N. J. A. Sloane

			22 is in the sequence because 22 = 10110_2 and '10110' has '110' as one of its substrings. - _Indranil Ghosh_, Feb 11 2017

Indranil Ghosh, Table of n, a(n) for n = 1..50000
Index entries for 2-automatic sequences.

Cf. A004748, A004750, A004751, A004752, A004753.

Select[Range[200],SequenceCount[IntegerDigits[#,2],{1,1,0}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2017 *)

is(n)=while(n>5,if(bitand(n,7)==6, return(1)); n>>=1); 0 \\ Charles R Greathouse IV, Feb 11 2017

i=j=0
while j<=500:
    if "110" in bin(i)[2:]:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 11 2017

a(n) ~ n. - Charles R Greathouse IV, Oct 23 2015

Offset corrected by Charles R Greathouse IV, Feb 11 2017

A004750 Numbers whose binary expansion contains the substring '011'.

Original entry on oeis.org

11, 19, 22, 23, 27, 35, 38, 39, 43, 44, 45, 46, 47, 51, 54, 55, 59, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 99, 102, 103, 107, 108, 109, 110, 111, 115, 118, 119, 123, 131, 134, 135, 139, 140, 141, 142, 143, 147, 150

Offset: 1

N. J. A. Sloane

			19 is in the sequence because 19 = 10011_2 and '10011' has '011' as one of its substrings. - _Indranil Ghosh_, Feb 11 2017

Indranil Ghosh, Table of n, a(n) for n = 1..32769 (terms 1..1001 from Harvey P. Dale)
Index entries for 2-automatic sequences.

Cf. A004749, A004751, A004752, A004753.

Select[Range[200],MemberQ[Partition[IntegerDigits[#,2],3,1],{0,1,1}]&] (* Harvey P. Dale, Apr 14 2014 *)
Select[Range[200],SequenceCount[IntegerDigits[#,2],{0,1,1}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 01 2021 *)

is(n)=while(n>10, if(bitand(n, 7)==3, return(1)); n>>=1); 0 \\ Charles R Greathouse IV, Feb 11 2017

i=j=0
while j<=100:
    if "011" in bin(i)[2:]:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 11 2017

a(n) ~ n. - Charles R Greathouse IV, Oct 23 2015

Offset corrected by Charles R Greathouse IV, Feb 11 2017

A004752 Binary expansion contains 010.

Original entry on oeis.org

10, 18, 20, 21, 26, 34, 36, 37, 40, 41, 42, 43, 50, 52, 53, 58, 66, 68, 69, 72, 73, 74, 75, 80, 81, 82, 83, 84, 85, 86, 87, 90, 98, 100, 101, 104, 105, 106, 107, 114, 116, 117, 122, 130, 132, 133, 136, 137, 138, 139, 144, 145, 146, 147, 148, 149, 150

Offset: 1

N. J. A. Sloane

			20 is in the sequence because, 20 = 10100_2 and '10100' has '010' as one of its substrings. - _Indranil Ghosh_, Feb 11 2017

Indranil Ghosh, Table of n, a(n) for n = 1..32769
Index entries for 2-automatic sequences.

Cf. A004748, A004749, A004750, A004751, A004753.

Select[Range@ 150, Length@ SequencePosition[IntegerDigits[#, 2], {0, 1, 0}] > 0 &] (* Version 10.1, or *)
Select[Range@ 150, MemberQ[Partition[IntegerDigits[#, 2], 3, 1], {0, 1, 0}] &] (* Michael De Vlieger, Feb 11 2017 *)

is(n)=while(n>9, if(bitand(n, 7)==2, return(1)); n>>=1); 0 \\ Charles R Greathouse IV, Feb 11 2017

i=j=0
while j<=100:
    if "010" in bin(i)[2:]:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 11 2017

a(n) ~ n. - Charles R Greathouse IV, Oct 23 2015

Offset corrected by Charles R Greathouse IV, Feb 11 2017

A004751 Binary expansion contains 001.

Original entry on oeis.org

9, 17, 18, 19, 25, 33, 34, 35, 36, 37, 38, 39, 41, 49, 50, 51, 57, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 89, 97, 98, 99, 100, 101, 102, 103, 105, 113, 114, 115, 121, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139

Offset: 1

N. J. A. Sloane

			19 is in the sequence because 19 = 10011_2 and '10011' has '001' as one of its substrings. - _Indranil Ghosh_, Feb 11 2017

Indranil Ghosh, Table of n, a(n) for n = 1..32769
Index entries for 2-automatic sequences.

Cf. A004748, A004749, A004750, A004752, A004753.

Select[Range@ 139, Length@ SequencePosition[IntegerDigits[#, 2], {0, 0, 1}] > 0 &] (* Version 10.1, or *)
Select[Range@ 139, MemberQ[Partition[IntegerDigits[#, 2], 3, 1], {0, 0, 1}] &] (* Michael De Vlieger, Feb 11 2017 *)
Select[Range[150],SequenceCount[IntegerDigits[#,2],{0,0,1}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 15 2018 *)

is(n)=while(n>8, if(bitand(n,7)==1, return(1)); n>>=1); 0 \\ Charles R Greathouse IV, Feb 11 2017

i=j=0
while j<=100:
    if "001" in bin(i)[2:]:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 11 2017

a(n) ~ n. - Charles R Greathouse IV, Oct 23 2015

Offset corrected by Charles R Greathouse IV, Feb 11 2017

A004779 Binary expansion contains 3 adjacent 0's.

Original entry on oeis.org

8, 16, 17, 24, 32, 33, 34, 35, 40, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 80, 81, 88, 96, 97, 98, 99, 104, 112, 113, 120, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 152, 160, 161, 162, 163, 168

Offset: 0

N. J. A. Sloane

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Index entries for 2-automatic sequences.

Complement of A003796.

Cf. A004753, A004781, A136037.

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

a(n) ~ n. - Charles R Greathouse IV, Oct 23 2015

A247875 Numbers that are even or whose binary expansions contain one or more pairs of adjacent zeros when odd.

Original entry on oeis.org

0, 2, 4, 6, 8, 9, 10, 12, 14, 16, 17, 18, 19, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86

Offset: 1

Reinhard Zumkeller, Sep 25 2014

nonn

Complement of A247648; union of A004753 and A005843.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A247648, A004753, A005843.

a247875 n = a247875_list !! (n-1)
a247875_list = filter (\x -> even x || f x) [0..] where
   f x = x > 0 && (x `mod` 4 == 0 || f (x `div` 2))

Module[{upto=100,odds},odds=Select[Range[1,upto,2], SequenceCount[ IntegerDigits[#,2],{0,0}]>0&];Sort[Join[odds,Range[0,upto,2]]]] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Aug 30 2015 *)

A247875_list = [n for n in range(10**3) if not n % 2 or '00' in bin(n)]
# Chai Wah Wu, Sep 26 2014

Definition edited by Chai Wah Wu, Sep 26 2014

Definition clarified by Harvey P. Dale, Aug 30 2015

A319302 Integers whose binary representation contains a consecutive string of zeros of prime length.

Original entry on oeis.org

4, 8, 9, 12, 17, 18, 19, 20, 24, 25, 28, 32, 34, 35, 36, 37, 38, 39, 40, 41, 44, 49, 50, 51, 52, 56, 57, 60, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 88, 89, 92, 96, 98, 99, 100, 101, 102, 103, 104, 105, 108, 113, 114, 115, 116, 120

Offset: 1

W. Zane Billings, Sep 16 2018

			81 = (1010001)_2 is a term because it contains a run of zeros of length 3, and 3 is a prime. 16 = (10000)_2 is not a term because it contains only a run of 4 zeros and 4 is not a prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004753, A318940.

Select[Range[120], AnyTrue[ Differences@ Flatten@ Position[ IntegerDigits[ 2*# + 1, 2], 1] - 1, PrimeQ] &] (* Giovanni Resta, Sep 17 2018 *)

is(n) = my(b=binary(n), i=0); for(k=1, #b, if(b[k]==0, i++); if(b[k]==1 || k==#b, if(ispseudoprime(i), return(1), i=0))); 0 \\ Felix Fröhlich, Sep 17 2018

from re import split
from sympy import isprime
A319302_list, n  = [], 1
while len(A319302_list) < 10000:
    for d in split('1+',bin(n)[2:]):
        if isprime(len(d)):
            A319302_list.append(n)
            break
    n += 1 # Chai Wah Wu, Oct 02 2018

More terms from Giovanni Resta, Sep 17 2018

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A003754 Numbers with no adjacent 0's in binary expansion.

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A328212 Lazy-Fibonacci-Niven numbers: numbers divisible by the number of terms in their lazy Fibonacci representation (A112310).

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A004748 Binary expansion contains 101.

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A004749 Numbers whose binary expansion contains the substring '110'.

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A004750 Numbers whose binary expansion contains the substring '011'.

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A004752 Binary expansion contains 010.

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A004751 Binary expansion contains 001.

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