A306556 Integers that appear as (unreduced) numerators of segment endpoints when a ternary Cantor set is created.

0, 1, 2, 3, 6, 7, 8, 9, 18, 19, 20, 21, 24, 25, 26, 27, 54, 55, 56, 57, 60, 61, 62, 63, 72, 73, 74, 75, 78, 79, 80, 81, 162, 163, 164, 165, 168, 169, 170, 171, 180, 181, 182, 183, 186, 187, 188, 189, 216, 217, 218, 219, 222, 223, 224, 225, 234, 235, 236, 237, 240, 241, 242, 243

Offset

1,3

Comments

Nonnegative integers whose ternary representation contains only digits 0 and 2 except for at most a single digit 1 that is followed only by 0's.

Nonnegative integers that can be written in base 3 using only 0's and 2's, allowing the use of the "decimal" point (.) and replacing ....10..0(.) by ....02..2(.)2222...

Note that fractions are not reduced.

List of integers in the closure of the ternary Cantor set under multiplication by 3. The closure is the union of the translated ternary Cantor sets spanning [a(1), a(2)], [a(3), a(4)], [a(5), a(6)], ... . - Peter Munn, Jul 09 2019

Links

Table of n, a(n) for n=1..64.

Georg Cantor, Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets), Part 5]. Mathematische Annalen (in German). (1883) 21: 545-591.

Paul du Bois-Reymond, Der Beweis des Fundamentalsatzes der Integralrechnung, Mathematische Annalen (in German), (1880), 16, footnote on p. 128.

Eric Weisstein's World of Mathematics, Cantor Set

Wikipedia, Cantor set

Index entries for 3-automatic sequences.

Formula

a(1)=0, a(2)=1;

a(2^n) = 3^(n-1) for n >= 1;

a(2^n+k) = 2*3^(n-1) + a(k) for 1 <= k <= 2^n.

From Peter Munn, Jul 09 2019: (Start)

a(2n-1) = A005823(n) = A191106(n)-1.

a(2n) = A191106(n) = A005823(n)+1.

a(2n-1) = (A055247(2n-1)-1)/3.

a(2n) = (A055247(2n) +1)/3.

a(2n-1) = (A191108(n)-1)/2.

a(2n) = (A191108(n)+1)/2.

(End)

Example

On 1st step we have [0,1/3] U [2/3,3/3] so we get a(1)=0, a(2)=1, a(3)=2, a(4)=3.
On 2nd step we have [0,1/9] U [2/9,3/9] U [6/9,7/9] U [8/9,9/9] so we get in addition a(5)=6, a(6)=7, a(7)=8, a(8)=9.

Prog

(PARI) A306556(n) = {sm=0; while(n>1, ex=floor(log(n)/log(2)); if(n-2^ex==0, sm=sm+3^(ex-1), sm=sm+2*3^(ex-1)); n=n-2^ex); return(sm)}
(PARI) a(n) = n--; fromdigits(binary(n>>1), 3)*2 + (n%2); \\ Kevin Ryde, Apr 23 2021

Crossrefs

Cf. A005823, A054591, A055247, A191106, A191108.

Sequence in context: A316156 A319737 A060132 * A257262 A293397 A059590

Adjacent sequences: A306553 A306554 A306555 * A306557 A306558 A306559

Keyword

nonn,frac,easy

Author

Dan Dima, Feb 23 2019

Status

approved