A005823 Numbers whose ternary expansion contains no 1's.

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

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn
• easy
• nice
• look
• base

Comments

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

References

• 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).

Links

• 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.

Crossrefs

Twice A005836.

Cf. A032924, A014263, A007089, A062756, A061392, A001196, A097252-A097262.

Cf. A088917 (characteristic function), A306556.

Programs

Maple

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

Mathematica

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

PARI

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

PARI

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

PARI

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

Python

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

Formula

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

Extensions

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.