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

%I A306556 #76 Aug 11 2025 08:09:10
%S A306556 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,
%T A306556 73,74,75,78,79,80,81,162,163,164,165,168,169,170,171,180,181,182,183,
%U A306556 186,187,188,189,216,217,218,219,222,223,224,225,234,235,236,237,240,241,242,243
%N A306556 Integers that appear as (unreduced) numerators of segment endpoints when a ternary Cantor set is created.
%C A306556 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.
%C A306556 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...
%C A306556 Note that fractions are not reduced.
%C A306556 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
%H A306556 Georg Cantor, <a href="https://gdz.sub.uni-goettingen.de/id/PPN235181684_0021">Über unendliche, lineare Punktmannigfaltigkeiten V</a>" [On infinite, linear point-manifolds (sets), Part 5]. Mathematische Annalen (in German). (1883) 21: 545-591.
%H A306556 Paul du Bois-Reymond, <a href="https://gdz.sub.uni-goettingen.de/id/PPN235181684_0016">Der Beweis des Fundamentalsatzes der Integralrechnung</a>, Mathematische Annalen (in German), (1880), 16, footnote on p. 128.
%H A306556 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CantorSet.html">Cantor Set</a>
%H A306556 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cantor_set">Cantor set</a>
%H A306556 <a href="/index/Ar#3-automatic">Index entries for 3-automatic sequences</a>.
%F A306556 a(1)=0, a(2)=1;
%F A306556 a(2^n) = 3^(n-1) for n >= 1;
%F A306556 a(2^n+k) = 2*3^(n-1) + a(k) for 1 <= k <= 2^n.
%F A306556 From _Peter Munn_, Jul 09 2019: (Start)
%F A306556 a(2n-1) = A005823(n) = A191106(n)-1.
%F A306556 a(2n)   = A191106(n) = A005823(n)+1.
%F A306556 a(2n-1) = (A055247(2n-1)-1)/3.
%F A306556 a(2n)   = (A055247(2n)  +1)/3.
%F A306556 a(2n-1) = (A191108(n)-1)/2.
%F A306556 a(2n)   = (A191108(n)+1)/2.
%F A306556 (End)
%e A306556 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.
%e A306556 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.
%o A306556 (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)}
%o A306556 (PARI) a(n) = n--; fromdigits(binary(n>>1),3)*2 + (n%2); \\ _Kevin Ryde_, Apr 23 2021
%Y A306556 Cf. A005823, A054591, A055247, A191106, A191108.
%K A306556 nonn,frac,easy
%O A306556 1,3
%A A306556 _Dan Dima_, Feb 23 2019