cp's OEIS Frontend

A378847 Smallest starting x which takes n tripling steps to reach the minimum of a cycle in the 3x-1 iteration.

%I A378847 #25 Jun 07 2025 13:06:31
%S A378847 1,3,15,13,9,37,25,33,45,57,145,97,65,87,159,165,225,273,391,261,647,
%T A378847 465,741,529,353,471,921,837,865,577,385,257,343,229,153,407,543,721,
%U A378847 481,321,855,1141,761,1015,677,903,1209,1605,2149,1433,1911,2529,3397,2265
%N A378847 Smallest starting x which takes n tripling steps to reach the minimum of a cycle in the 3x-1 iteration.
%C A378847 Each step is x -> 3x-1 if x odd, or x -> x/2 if x even (A001281) and here only the tripling steps 3x-1 are counted.
%C A378847 The number of tripling steps is A378833(x) so that a(n) = x is the smallest x for which A378833(x) = n.
%C A378847 All terms are odd since any even x takes a first step to x/2 which is a smaller start for the same number of tripling steps.
%C A378847 a(n) >= L(n) = (2*a(n-1) + 1)/3 is a lower bound since a(n) = x must at least have a first step 3x-1 and halve to (3x-1)/2, then n-1 further tripling steps, so (3x-1)/2 >= a(n-1).
%C A378847 Equality a(n) = L(n) occurs iff L(n) is an integer and not a cycle minimum.
%C A378847 A large upper bound for n>=1, showing a(n) always exists, is a(n) <= U(n) = (4^(3^n) - 1)*2^n/3^n + 1, since U(n) is a candidate for a(n) by taking n steps of (3x-1)/2 to reach 4^(3^n) which is a power of 2.
%C A378847 Tighter upper bounds on a(n) can be found by taking predecessor steps back from a(n-c) seeking c tripling steps to reach a(n-c) if that's possible (which for instance it's not if a(n-c) == 0 (mod 3)).
%C A378847 Such predecessors are candidates for a(n), but the actual a(n) might have a trajectory which does not go through any previous a(n-c).
%H A378847 Kevin Ryde, <a href="/A378847/b378847.txt">Table of n, a(n) for n = 0..625</a>
%H A378847 Kevin Ryde, <a href="/A378845/a378845.c.txt">C Code</a> (set count type TRIPLE)
%e A378847 For n=4, a(4) = 9 has 4 tripling steps on its way to 5 which is the minimum of a cycle:
%e A378847   9 -> 26 -> 13 -> 38 -> 19 -> 56 -> 28 -> 14 -> 7 -> 20 -> 10 -> 5
%e A378847     ^            ^           ^                      ^
%e A378847 This a(4) = 9 is an example where a(n) is at its lower bound L(n), in this case a(3) = 13 has L(4) = (2*a(3)+1)/3 = 9 which is an integer and not a cycle minimum.
%o A378847 (C) /* See links. */
%Y A378847 Cf. A001281 (step), A378833 (number of triplings).
%Y A378847 Cf. A378845 (with all steps), A378846 (with halving steps).
%K A378847 nonn
%O A378847 0,2
%A A378847 _Kevin Ryde_, Dec 15 2024