cp's OEIS Frontend

A092893 Smallest starting value in a Collatz '3x+1' sequence such that the sequence contains exactly n tripling steps.

%I A092893 #37 Jun 20 2025 13:44:17
%S A092893 1,5,3,17,11,7,9,25,33,43,57,39,105,135,185,123,169,219,159,379,283,
%T A092893 377,251,167,111,297,395,263,175,233,155,103,137,91,121,161,107,71,47,
%U A092893 31,41,27,73,97,129,171,231,313,411,543,731,487,327,859,1145,763,1017,1351
%N A092893 Smallest starting value in a Collatz '3x+1' sequence such that the sequence contains exactly n tripling steps.
%C A092893 First occurrence of n in A006667.
%C A092893 These are the odd (primitive) terms in A129304. - _T. D. Noe_, Apr 09 2007
%C A092893 Except for a(1) = 5, all values are congruent {1, 3, 7} (mod 8). Reason: If n is 5 (mod 8) then the Collatz trajectory starting with m = (n - 1)/4 contains the same number of tripling steps, because n = 4m + 1 and the Collatz 3x + 1 step results in 3*(4m + 1) + 1 = 12m + 4 which gets reduced by halving to 3m + 1, without changing the number of tripling steps. - _Ralf Stephan_, Jun 19 2025
%H A092893 T. D. Noe, <a href="/A092893/b092893.txt">Table of n, a(n) for n=0..300</a>
%H A092893 Jeffrey R. Goodwin, <a href="http://arxiv.org/abs/1504.03040">The 3x+1 Problem and Integer Representations</a>, Arxiv preprint arXiv:1504.03040 [math.NT], 2015.
%H A092893 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A092893 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A092893 a(4)=11 because the Collatz sequence 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 is the first sequence containing 4 tripling steps.
%t A092893 a[n_]:=Length[Select[NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &],OddQ]]; Table[i=1; While[a[i]!=n,i=i+2]; i,{n,58}] (* _Jayanta Basu_, May 27 2013 *)
%Y A092893 Cf. A006667, A033491, A092892, A087228.
%Y A092893 Row n=1 of A354236.
%K A092893 nonn
%O A092893 0,2
%A A092893 _Hugo Pfoertner_, Mar 11 2004