A026179 - OEIS

%I #55 Feb 16 2025 01:11:04

%S 1,2,5,6,8,11,14,15,17,18,20,23,24,26,29,32,33,35,38,41,42,44,45,47,

%T 50,51,53,54,56,59,60,62,65,68,69,71,72,74,77,78,80,83,86,87,89,92,95,

%U 96,98,99,101,104,105,107,110,113,114,116,119

%N Numbers k such that A026177(j) < A026177(k) for all j < k.

%C After first term, these are the numbers of the form (3i+2)*3^j, where i >= 0, j >= 0. - _Clark Kimberling_, Oct 19 2016

%C Old conjecture: (a(n)) = complement of A026225 after removal of the initial 1 here. [Note that following the proof of the form of its terms, A026225 has been renamed accordingly. - _Peter Munn_, Mar 24 2022]

%C The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Apr 03 2022

%H Amiram Eldar, <a href="/A026179/b026179.txt">Table of n, a(n) for n = 1..10000</a>

%H F. M. Dekking, <a href="https://arxiv.org/abs/2001.08915">Permutations of N generated by left-right filling algorithms</a>, arXiv:2001.08915 [math.CO], 2020, see R_opos in section 2.3.

%H Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "TurnRight".

%H <a href="/index/Ar#3-automatic">Index entries for 3-automatic sequences</a>.

%F Let the sequence 1, 0, 1, 1, 0, 0, 1, 0, 1, ... (A137893) be defined as the fixed point of the morphism 1->101 and 0->100, starting from a(1)=1. The indices of 0 are 2, 5, 6, 8, 11, 14, 17, 18, ... (this sequence with first term omitted). - _Philippe Deléham_, Jun 27 2006

%t a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, 160}]

%t p[b_, d_] := Flatten[Position[a[b], d]]

%t p[3, 1] (* A026225 *)

%t p[3, 2] (* A026179 without initial 1 *)

%t (* _Clark Kimberling_, Oct 19 2016 *)

%o (PARI) a(n) = { if(n>1, n=2*n-2; my(v=digits(n,3));

%o   for(i=1,#v, if(v[i]==1, n++;

%o     forstep(j=#v,i,-1, if(v[j]++>2,v[j]=0,break)))));

%o   n; } \\ _Kevin Ryde_, Apr 23 2021

%o (Python)

%o from sympy import integer_log

%o def A026179(n):

%o     def f(x): return n-1+x-sum(((x//3**i)-2)//3+1 for i in range(integer_log(x,3)[0]+1))

%o     m, k = n, f(n)

%o     while m != k: m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Feb 15 2025

%Y Cf. A026177, A026180, A026225.

%Y Cf. A080846 (characteristic function except for 1), A137893.

%K nonn,changed

%O 1,2

%A _Clark Kimberling_