A061392 -id:A061392 - cp's OEIS frontend

A061393 Number of appearances of n in sequence defined by b(k) = b(floor(k/3)) + b(ceiling(k/3)) with b(0)=0 and b(1)=1, i.e., in A061392.

1, 2, 4, 2, 10, 2, 4, 2, 28, 2, 4, 2, 10, 2, 4, 2, 82, 2, 4, 2, 10, 2, 4, 2, 28, 2, 4, 2, 10, 2, 4, 2, 244, 2, 4, 2, 10, 2, 4, 2, 28, 2, 4, 2, 10, 2, 4, 2, 82, 2, 4, 2, 10, 2, 4, 2, 28, 2, 4, 2, 10, 2, 4, 2, 730, 2, 4, 2, 10, 2, 4, 2, 28, 2, 4, 2, 10, 2, 4, 2, 82, 2, 4, 2, 10, 2, 4, 2, 28, 2, 4, 2

Offset: 0

Henry Bottomley, Apr 30 2001

nonn

In the binary expansion of n, delete everything left of the rightmost 1 bit, then interpret as ternary and add 1. - Ralf Stephan, Aug 22 2013

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Gilleland, Some Self-Similar Integer Sequences
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

Cf. A061392.

A061393(n) = if(!n,1,(1+3^valuation(n,2))); \\ Antti Karttunen, Sep 30 2018

a(n) = A034472(A007814(n)) for n > 0.
a(2n) = 3a(n)-2; a(2n+1) = 2.
G.f.: 1/(1-x) + Sum_{k>=0} 3^k*x^2^k/(1 - x^2^(k+1)). - Ralf Stephan, Jun 13 2003

A005823 Numbers whose ternary expansion contains no 1's.

Original entry on oeis.org

0, 2, 6, 8, 18, 20, 24, 26, 54, 56, 60, 62, 72, 74, 78, 80, 162, 164, 168, 170, 180, 182, 186, 188, 216, 218, 222, 224, 234, 236, 240, 242, 486, 488, 492, 494, 504, 506, 510, 512, 540, 542, 546, 548, 558, 560, 564, 566, 648, 650, 654, 656, 666, 668, 672, 674

Offset: 1

N. J. A. Sloane, Jeffrey Shallit

The set of real numbers between 0 and 1 that contain no 1's in their ternary expansion is the well-known Cantor set with Hausdorff dimension log 2 / log 3.
Complement of A081606. - Reinhard Zumkeller, Mar 23 2003
Numbers k such that the k-th Apery number is congruent to 1 (mod 3) (cf. A005258). - Benoit Cloitre, Nov 30 2003
Numbers k such that the k-th central Delannoy number is congruent to 1 (mod 3) (cf. A001850). - Benoit Cloitre, Nov 30 2003
Numbers k such that there exists a permutation p_1, ..., p_k of 1, ..., k such that i + p_i is a power of 3 for every i. - Ray Chandler, Aug 03 2004
Subsequence of A125292. - Reinhard Zumkeller, Nov 26 2006
The first 2^n terms of the sequence could be obtained using the Cantor process for the segment [0,3^n-1]. E.g., for n=2 we have [0,{1},2,{3,4,5},6,{7},8]. The numbers outside of braces are the first 4 terms of the sequence. Therefore the terms of the sequence could be called "Cantor's numbers". - Vladimir Shevelev, Jun 13 2008
Mahler proved that positive a(n) is never a square. - Michel Marcus, Nov 12 2012
Define t: Z -> P(R) so that t(k) is the translated Cantor ternary set spanning [k, k+1], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3. - Peter Munn, Oct 30 2019

K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Winston de Greef, Table of n, a(n) for n = 1..16384 (first 1024 terms from T. D. Noe)
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98, No. 2 (1992), pp. 163-197.
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98, No. 2 (1992), pp. 163-197.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Sajed Haque, Chapter 3.4 of Discriminators of Integer Sequences, 2017, See p. 45.
Sajed Haque and Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 147-160.
Kurt Mahler, The representation of squares to the base 3, Acta Arith., Vol. 53, Issue 1 (1989), pp. 99-106.
M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., Vol. 65, No. 2 (1989), pp. 213-220.
Eric Weisstein's World of Mathematics, Cantor Set.
Index entries for 3-automatic sequences.

Twice A005836.
Cf. A032924, A014263, A007089, A062756, A061392, A001196, A097252-A097262.
Cf. A088917 (characteristic function), A306556.

a:= proc(n) option remember;
      `if`(n=1, 0, `if`(irem (n, 2, 'q')=0, 3*a(q)+2, 3*a(q+1)))
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 19 2012

Select[ Range[ 0, 729 ], (Count[ IntegerDigits[ #, 3 ], 1 ]==0)& ]
Select[Range[0,700],DigitCount[#,3,1]==0&] (* Harvey P. Dale, Mar 12 2016 *)

is(n)=while(n,if(n%3==1,return(0),n\=3));1 \\ Charles R Greathouse IV, Apr 20 2012

a(n)=n=binary(n-1);sum(i=1,#n,2*n[i]*3^(#n-i)) \\ Charles R Greathouse IV, Apr 20 2012

a(n)=2*fromdigits(binary(n-1),3) \\ Charles R Greathouse IV, Aug 24 2016

def A005823(n):
    return 2*int(format(n-1,'b'),3) # Chai Wah Wu, Jan 04 2015

a(n) = 2 * A005836(n).
a(2n) = 3*a(n)+2, a(2n+1) = 3*a(n+1), a(1) = 0.
a(n) = Sum_{k = 1..n} 1 + 3^A007814(k). - Philippe Deléham, Jul 09 2005
A125291(a(n)) = 1 for n>0. - Reinhard Zumkeller, Nov 26 2006
From Reinhard Zumkeller, Mar 02 2008: (Start)
A062756(a(n)) = 0.
If the offset were changed to zero, then: a(0) = 0, a(n+1) = f(a(n)+1, a(n)+1) where f(x, y) = if x < 3 and x <> 1 then y else if x mod 3 = 1 then f(y+1, y+1) else f(floor(x/3), y). (End)
G.f. g(x) satisfies g(x) = 3*g(x^2)*(1+1/x) + 2*x^2/(1-x^2). - Robert Israel, Jan 04 2015
Sum_{n>=2} 1/a(n) = 1.341426555483087715426958452292349687410838545707857407585878304836140592352... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

More terms from Sascha Kurz, Mar 24 2002

Offset corrected by N. J. A. Sloane, Mar 02 2008. This may require some of the formulas to be adjusted.

A081608 Number of numbers <= n having no 0 in their ternary representation.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 17, 18, 18, 18, 18, 18, 19, 20, 20, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 24, 24, 25, 26, 26, 26, 26

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A081607(n) = n+1.

Winston de Greef, Table of n, a(n) for n = 0..16383

Cf. A007089, A077267, A061392, A081611.

Accumulate[Table[If[DigitCount[n,3,0]==0,1,0],{n,0,80}]] (* Harvey P. Dale, Oct 21 2024 *)

first(n)=my(s,t); vector(n,k, t=Set(digits(k,3)); s+=!!t[1]) \\ Charles R Greathouse IV, Sep 02 2015

A081611 Number of numbers <= n having no 2 in their ternary representation.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) is also the size of the subset of [1..n] when numbers are added greedily so as to not contain a 3-term arithmetic progression, i.e., according to A003278: a(n) = the largest k such that A003278(k) <= n. (Cf. A003002, the size of the optimal (largest) 3-free subset of [1..n].) - Shreevatsa R, Oct 19 2009

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

Cf. A003278, A007089, A039966, A081603, A081608, A061392, A081610.

a081611 n = a081611_list !! n
a081611_list = scanl1 (+) a039966_list
-- Reinhard Zumkeller, Jan 28 2012

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (&*[1+x^(3^k): k in [0..5]])/(1-x) )); // G. C. Greubel, Mar 29 2019

CoefficientList[Series[Product[1+x^(3^k), {k,0,5}]/(1-x), {x,0,100}], x] (* G. C. Greubel, Mar 29 2019 *)

{a(n)=local(A,m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*(1+x+x^2)*subst(A,x,x^3)); polcoeff(A,n))} /* Michael Somos, Aug 31 2006 */

first(n)=my(s,t); vector(n,k,t=Set(digits(k,3)); s+=t[#t]<2) \\ Charles R Greathouse IV, Sep 02 2015

my(x='x+O('x^100)); Vec(prod(k=0,5,1+x^(3^k))/(1-x)) \\ G. C. Greubel, Mar 29 2019

from gmpy2 import digits
def A081611(n):
    l = (s:=digits(n,3)).find('2')
    if l >= 0: s = s[:l]+'1'*(len(s)-l)
    return int(s,2)+1 # Chai Wah Wu, Dec 05 2024

(product(1+x^(3^k) for k in (0..5))/(1-x)).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Mar 29 2019

a(n) + A081610(n) = n+1.
From Michael Somos, Aug 31 2006: (Start)
G.f. A(x) satisfies A(x)=(1+x)*(1+x+x^2)*A(x^3).
G.f.: (1/(1-x))*Product_{k>=0} (1+x^(3^k)).
a(n) = a(n-1) + A039966(n). (End)

A081609 Number of numbers <= n having at least one 1 in their ternary representation.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 47, 48, 49, 50, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A061392(n) = n+1.

Cf. A007089, A062756, A081607, A081610, A061392.

Partial sums of A316829.

has(n)=!!setsearch(Set(digits(n,3)),1)
first(n)=my(s); vector(n,k,s+=has(k)) \\ Charles R Greathouse IV, Sep 02 2015

A326538 a(n) is the numerator of the image of 1/n by the Cantor staircase function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 3, 3, 3, 23, 1, 5, 5, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 7, 7

Offset: 1

Rémy Sigrist, Jul 12 2019

The Cantor staircase function, say c, maps rational numbers in the interval [0..1] to rational numbers in the interval [0..1], hence this sequence is well defined.

For any n > 0, the binary expansion of c(1/n) is terminating (and A326539(n) is a power of 2) iff the ternary expansion of 1/n is terminating or contains a digit 1.

			The first terms, alongside c(1/n) and the ternary and binary representation of 1/n and c(1/n), respectively, with periodic part in parentheses, are:
  n   a(n)  c(1/n)  ter(1/n)                bin(c(1/n))
  --  ----  ------  ----------------------  -----------
   1     1       1  1.(0)                   1.(0)
   2     1     1/2  0.(1)                   0.1(0)
   3     1     1/2  0.1(0)                  0.1(0)
   4     1     1/3  0.(02)                  0.(01)
   5     1     1/4  0.(0121)                0.01(0)
   6     1     1/4  0.0(1)                  0.01(0)
   7     1     1/4  0.(010212)              0.01(0)
   8     1     1/4  0.(01)                  0.01(0)
   9     1     1/4  0.01(0)                 0.01(0)
  10     1     1/5  0.(0022)                0.(0011)
  11     3    3/16  0.(00211)               0.0011(0)
  12     1     1/6  0.0(02)                 0.0(01)
  13     1     1/7  0.(002)                 0.(001)
  14     1     1/8  0.(001221)              0.001(0)
  15     1     1/8  0.0(0121)               0.001(0)
  16     1     1/8  0.(0012)                0.001(0)
  17     1     1/8  0.(0011202122110201)    0.001(0)
  18     1     1/8  0.00(1)                 0.001(0)
  19     1     1/8  0.(001102100221120122)  0.001(0)
  20     1     1/8  0.(0011)                0.001(0)

Rémy Sigrist, Table of n, a(n) for n = 1..6561
Rémy Sigrist, PARI program for A326538
Wikipedia, Cantor function

See A326539 for the corresponding denominators.

Cf. A061392.

PARI
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See Links section.
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A340619 n appears A006519(n) times.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 19, 20, 20, 20, 20, 21, 22, 22, 23, 24, 24, 24, 24, 24, 24, 24, 24, 25, 26, 26

Offset: 1

Rémy Sigrist, Jan 13 2021

This sequence has similarities with the Cantor staircase function.
This sequence can be seen as an irregular table where the n-th row contains A006519(n) times the value n.
For any k > 1, the set of points { (n, a(n)), n = 1..A006520(2^k-1) } is symmetric; for example, for k = 3, we have the following configuration:
      a(n)
       ^
       |           *
       |         **
       |        *
       |    ****
       |   *
       | **
       |*
       +-------------> n

			The first rows, alongside A006519(n), are:
    n | n-th row               | A006519(n)
   ---+------------------------+-----------
    1 | 1                      |          1
    2 | 2, 2                   |          2
    3 | 3                      |          1
    4 | 4, 4, 4, 4             |          4
    5 | 5                      |          1
    6 | 6, 6                   |          2
    7 | 7                      |          1
    8 | 8, 8, 8, 8, 8, 8, 8, 8 |          8
    9 | 9                      |          1
   10 | 10, 10                 |          2

Rémy Sigrist, Table of n, a(n) for n = 1..11264
Wikipedia, Cantor function

See A061392 and A340500 for similar sequences.

Cf. A006519, A006520, A046699.

A340619[n_] := Array[n &, Table[BitAnd[BitNot[i - 1], i], {i, 1, n}][[n]]];
Table[A340619[n], {n, 1, 26}] // Flatten (* Robert P. P. McKone, Jan 19 2021 *)

concat(apply(v -> vector(2^valuation(v,2), k, v), [1..26]))

a(n) = my(ret=0); forstep(k=logint(n,2),0,-1, if(n > k<<(k-1), ret+=1<Kevin Ryde, Jan 18 2021

a(A006520(n)) = n.
a(A006520(n)+1) = n+1.
a(n) + a(A006520(2^k-1) + 1 - n) = 2^k for any k > 0 and n = 1..A006520(2^k-1).
a(n) = 2^k + (a(r) if r>0), where k such that k*2^(k-1) < n <= (k+1)*2^k and r = n - (k+2)*2^(k-1). - Kevin Ryde, Jan 18 2021

cp's OEIS Frontend

A061393 Number of appearances of n in sequence defined by b(k) = b(floor(k/3)) + b(ceiling(k/3)) with b(0)=0 and b(1)=1, i.e., in A061392.

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A005823 Numbers whose ternary expansion contains no 1's.

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A081608 Number of numbers <= n having no 0 in their ternary representation.

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A081611 Number of numbers <= n having no 2 in their ternary representation.

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A081609 Number of numbers <= n having at least one 1 in their ternary representation.

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A326538 a(n) is the numerator of the image of 1/n by the Cantor staircase function.

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A340619 n appears A006519(n) times.

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