cp's OEIS Frontend

A378845 Smallest starting x which takes n steps to reach the minimum of a cycle in the 3x-1 iteration.

%I A378845 #15 Dec 27 2024 00:43:52
%S A378845 1,2,4,7,3,6,11,19,21,13,26,9,18,35,37,73,25,49,98,33,66,131,45,90,
%T A378845 175,127,117,85,149,57,113,199,209,133,265,89,177,65,119,237,87,159,
%U A378845 165,329,231,225,439,309,293,585,377,391,273,261,521,1042,671,695,485
%N A378845 Smallest starting x which takes n steps to reach the minimum of a cycle in the 3x-1 iteration.
%C A378845 Each step is x -> 3x-1 if x odd, or x -> x/2 if x even (A001281).
%C A378845 The number of steps is A135730(x) so that a(n) = x is the smallest x for which A135730(x) = n.
%C A378845 a(n) <= 2*a(n-1) since x = 2*a(n-1) is a candidate for a(n) by first step x -> x/2.
%C A378845 Even terms are always a(n) = 2*a(n-1) since any smaller even a(n) would imply a smaller a(n-1) after first step x -> x/2.
%C A378845 No term is of the form 12*k+4, since its first step to 6*k+2 is also where the first step from 2*k+1 goes and the latter is a smaller start.
%C A378845 a(n) >= (a(n-1) + 1)/3 is a lower bound since a(n) = x must at least have a first step 3x-1 which reaches somewhere with n-1 further steps, so 3x-1 >= a(n-1).
%C A378845 Equality a(n) = (a(n-1) + 1)/3 = x occurs iff that x is an odd integer and not a cycle minimum, so its first step is to 3x-1 = a(n-1) (as for example at n=11).
%H A378845 Kevin Ryde, <a href="/A378845/b378845.txt">Table of n, a(n) for n = 0..1435</a>
%H A378845 Kevin Ryde, <a href="/A378845/a378845.c.txt">C Code</a>
%o A378845 (C) /* See links. */
%Y A378845 Cf. A001281 (step), A135730 (number of steps).
%Y A378845 Cf. A378846 (with halving steps), A378847 (with tripling steps).
%Y A378845 Cf. A033491 (in 3x+1).
%K A378845 nonn
%O A378845 0,2
%A A378845 _Kevin Ryde_, Dec 09 2024