A144795 -id:A144795 - cp's OEIS frontend

A175059 a(n) = decimal equivalent of {A144795(n) written in binary, and each run of 1's reduced in length by one digit}.

1, 2, 3, 4, 6, 7, 8, 5, 12, 14, 15, 16, 9, 10, 11, 24, 13, 28, 30, 31, 32, 17, 18, 19, 20, 22, 23, 48, 25, 26, 27, 56, 29, 60, 62, 63, 64, 33, 34, 35, 36, 38, 39, 40, 21, 44, 46, 47, 96, 49, 50, 51, 52, 54, 55, 112, 57, 58, 59, 120, 61, 124, 126, 127, 128, 65, 66, 67, 68, 70

Offset: 1

Leroy Quet, Dec 08 2009

This is a permutation of the positive integers. Sequence A175060 is its inverse permutation.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A144795, A175055, A175056, A175057, A175058, A175060.

Map[FromDigits[#, 2] &@ Flatten@ # &, Select[Array[Split@ IntegerDigits[#, 2] &, 400], FreeQ[#, {1}] &] /. w_List /; First@ w == 1 :> Most@ w] (* Michael De Vlieger, Sep 03 2017 *)

Extended by Ray Chandler, Dec 18 2009

A173022 Number of numbers <= n whose binary representation is without isolated ones.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21

Offset: 0

Reinhard Zumkeller, Feb 07 2010

			a(20) = #{0,3,6,7,12,14,15} = #{0,11,110,111,1100,1110,1111} = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A005251, A007088, A144795, A173021, A173023.

Accumulate[Array[Boole[FreeQ[Split[IntegerDigits[#, 2]], {1}]] &, 100, 0]] (* Paolo Xausa, Oct 15 2024 *)

a(A144795(n+1)) = a(A144795(n)) + 1.
a(2^n - 1) = A005251(n+2).
A173021(n) <= a(n) <= A173023(n).

A173024 Numbers having neither isolated ones nor isolated double ones in their binary representations.

Original entry on oeis.org

0, 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 119, 120, 124, 126, 127, 224, 231, 238, 239, 240, 247, 248, 252, 254, 255, 448, 455, 462, 463, 476, 478, 479, 480, 487, 494, 495, 496, 503, 504, 508, 510, 511, 896, 903, 910, 911, 924, 926, 927, 952, 956, 958, 959

Offset: 1

Reinhard Zumkeller, Feb 07 2010

Intersection of A144795 and A173025.
If m is a term then 2*m is also a term.
If m is an odd term then 2*m+1 is also a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005251, A144795, A173021.

nioQ[n_]:=Min[Length/@Select[Split[IntegerDigits[n,2]],FreeQ[#,0]&]]>2; Select[ Range[ 0,1000],nioQ] (* Harvey P. Dale, Jan 12 2023 *)

A173021(a(n+1)) = A173021(a(n)) + 1.

A173025 Numbers whose binary representation contains no isolated digits "11".

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 9, 10, 14, 15, 16, 17, 18, 20, 21, 23, 28, 29, 30, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 46, 47, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 71, 72, 73, 74, 78, 79, 80, 81, 82, 84, 85, 87, 92, 93, 94, 95, 112, 113, 114, 116, 117, 119, 120, 121, 122

Offset: 1

Reinhard Zumkeller, Feb 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007088, A144795, A173021.

A173024 is a subsequence.

Select[Range[0, 150], FreeQ[Split[IntegerDigits[#, 2]], {1, 1}] &] (* Paolo Xausa, Oct 15 2024 *)

A173021(a(n+1)) = A173021(a(n)) + 1.

A144794 A positive integer n is included if every 0 in binary n is next to at least one other 0.

Original entry on oeis.org

4, 8, 9, 12, 16, 17, 19, 24, 25, 28, 32, 33, 35, 36, 39, 48, 49, 51, 56, 57, 60, 64, 65, 67, 68, 71, 72, 73, 76, 79, 96, 97, 99, 100, 103, 112, 113, 115, 120, 121, 124, 128, 129, 131, 132, 135, 136, 137, 140, 143, 144, 145, 147, 152, 153, 156, 159, 192, 193, 195, 196

Offset: 1

Leroy Quet, Sep 21 2008

n is included if A144789(n) >= 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A144789, A144795.

A007814 := proc(n) local nshf,a ; a := 0 ; nshf := n ; while nshf mod 2 = 0 do nshf := nshf/2 ; a := a+1 ; od: a ; end: A144789 := proc(n) option remember ; local lp2,lp2sh,bind ; bind := convert(n,base,2) ; if add(i,i=bind) = nops(bind) then RETURN(0) ; fi; lp2 := A007814(n) ; if lp2 = 0 then A144789(floor(n/2)) ; else lp2sh := A144789(n/2^lp2) ; if lp2sh = 0 then lp2 ; else min(lp2,lp2sh) ; fi; fi; end: isA144794 := proc(n) RETURN(A144789(n) >= 2) ; end: for n from 3 to 400 do if isA144794(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 29 2008

Select[Range@ 200, And[Union@ # != {1}, AllTrue[Map[Length, Select[Split@ #, First@ # == 0 &]], # > 1 &]] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Aug 20 2017 *)

Extended by R. J. Mathar, Sep 29 2008

A336880 Numbers with decimal expansion d_1, ..., d_w such that for any k in 1..w there is some m in 1..w such that d_k = d_m = abs(k - m).

Original entry on oeis.org

0, 11, 110, 111, 202, 1100, 1110, 1111, 2020, 2222, 3003, 3113, 11000, 11011, 11100, 11110, 11111, 11202, 20200, 20202, 20211, 22220, 22222, 23203, 30030, 30232, 31130, 33033, 40004, 40114, 41104, 41114, 42024, 110000, 110011, 110110, 110111, 110202, 111000

Offset: 1

Rémy Sigrist, Aug 06 2020

This sequence has similarities with A336668.
All repunits (A002275) belong to this sequence.
The concatenation of two terms is also a term.
The digit reversal (A004086) of a term is also a term.
For any d in 1..9, d * (1 + 10^d) is the first term containing the digit d.

			Regarding 30232:
- the first digit 3 is 3 positions away from the second digit 3 and vice versa,
- the digit 0 matches itself,
- the first digit 2 is 2 positions away from the second digit 2 and vice versa,
- so 30232 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002275, A144795 (binary analog), A336668.

is(n, base=10) = { my (d=digits(n, base)); for (k=1, #d, if ((k-d[k]<1 || d[k-d[k]]!=d[k]) && (k+d[k]>#d || d[k+d[k]]!=d[k]), return (0)));
return (1) }

A377169 Nonnegative integers containing isolated ones in their binary representation.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 57, 58, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Paolo Xausa, Oct 18 2024

A 1 is isolated if it's not adjacent to another 1.

			19 is a term because 19 = 10011_2 contains one isolated 1.
74 is a term because all ones in 74 = 1001010_2 are isolated.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Complement of A144795.

Cf. A004752, A007088, A377167.

Select[Range[150], MemberQ[Split[IntegerDigits[#, 2]], {1}] &]

A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

Original entry on oeis.org

1, 3, 2, 6, 4, 7, 8, 12, 5, 11, 9, 14, 16, 15, 13, 19, 21, 22, 10, 23, 17, 24, 18, 27, 29, 30, 26, 35, 52, 38, 25, 31, 32, 28, 33, 47, 34, 46, 20, 39, 40, 44, 36, 43, 37, 48, 41, 75, 53, 59, 61, 54, 57, 55, 58, 60, 64, 51, 65, 56, 66, 62, 50, 71, 45, 78, 42

Offset: 1

Rémy Sigrist, Jun 28 2017

A144795 gives the numbers without isolated 1's in base-2 representation.
This sequence is conjectured to be a permutation of the natural numbers.
This sequence has similarities with A269361: here we require that the product of two consecutive terms has no isolated 1, there the product of two consecutive terms has only isolated 1's, in base-2 representation.
For any k > 0:
- a(2*k-1) belongs to A091072,
- a(2*k) belongs to A091067.

			The first terms, alongside a(n)*a(n+1) in binary, are:
n       a(n)    a(n)*a(n+1) in binary
--      ----    ---------------------
1       1       11
2       3       110
3       2       1100
4       6       11000
5       4       11100
6       7       111000
7       8       1100000
8       12      111100
9       5       110111
10      11      1100011
11      9       1111110
12      14      11100000
13      16      11110000
14      15      11000011
15      13      11110111
16      19      110001111
17      21      111001110
18      22      11011100
19      10      11100110
20      23      110000111

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A289194

Cf. A091067, A091072, A144795, A269361.

a = {1}; Do[k = 1; While[Nand[! MemberQ[a, k], ! MemberQ[Length /@ DeleteCases[Split[IntegerDigits[k Last[a], 2]], s_ /; First@ s == 0], 1]], k++]; AppendTo[a, k], {n, 2, 67}]; a (* Michael De Vlieger, Jun 29 2017 *)

cp's OEIS Frontend

A175059 a(n) = decimal equivalent of {A144795(n) written in binary, and each run of 1's reduced in length by one digit}.

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A173022 Number of numbers <= n whose binary representation is without isolated ones.

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A173024 Numbers having neither isolated ones nor isolated double ones in their binary representations.

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A173025 Numbers whose binary representation contains no isolated digits "11".

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A144794 A positive integer n is included if every 0 in binary n is next to at least one other 0.

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Maple

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A336880 Numbers with decimal expansion d_1, ..., d_w such that for any k in 1..w there is some m in 1..w such that d_k = d_m = abs(k - m).

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PARI

A377169 Nonnegative integers containing isolated ones in their binary representation.

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Mathematica

A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

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