%I
%S 0,2,1,1,2,3,2,1,0,0,1,3,2,1,1,6,3,5,2,4,4,0,3,3,8,8,2,7,1,4,0,3,6,3,
%T 1,2,5,10,1,4,10,7,1,9,3,9,3,8,0,8,2,2,5,7,0,7,7,7,1,4,1,2,6,6,6,9,3,
%U 1,2,5,5,5,5,8,2,2,1,10,4,16,4,4,4,4,9,6,1,9,3,15,3,3,3,6,3,3,2,8,8,2,8,14
%N Start at 2n+1, iterate the map x > A337349(x); sequence gives the number of iterations to resulting cycle or 1 if the trajectory never cycles.
%C Iteration: multiply by 3 and add 1 and divide out any power of 2; then multiply by 3 and subtract 1 and divide out any power of 2.
%C When a(x) is iterated, what are the limit cycles? Are there any besides {1} and {17 > 19 > 43 > 97 > 109 > 61}?
%H Ray Chandler, <a href="/A122563/b122563.txt">Table of n, a(n) for n = 0..10000</a>
%e The iteration for n=13 is 27>61>17>19>43>97>109>61>... and a(13)=1 step was needed to enter the cycle (at 61).
%e The iteration for n=30 is 61> 17>19>43>97>109>61>> and the cycle was already entered at the start, so a(30)=0.
%p A122563 := proc(n)
%p local cyc,itr,x ;
%p cyc := [] ;
%p x := 2*n+1 ;
%p while true do
%p cyc := [op(cyc),x] ;
%p x := A337349(x) ;
%p if x in cyc then
%p break ;
%p end if;
%p end do:
%p member(x,cyc,'itr') ;
%p itr 1 ;
%p end proc:
%p seq(A122563(n),n=0..101) ; # _R. J. Mathar_, Aug 26 2020
%t nextx[x_Integer] := Block[{a = x}, a = 3 a + 1; While[EvenQ@a, a /= 2]; a = 3 a  1; While[EvenQ@a, a /= 2]; a]; f[n_] := Length@NestWhileList[nextx, n, FreeQ[{1, 17, 19, 43, 97, 109, 61}, #] &]  1; Table[f[2 n + 1], {n, 0, 101}] (* original program from author corrected as suggested by _William P. Orrick_, _Ray Chandler_, Aug 28 2020 *)
%Y Cf. A102421, A102423, A337349.
%K nonn,look
%O 0,2
%A _Robert G. Wilson v_, based on email from Dan Asimov (dasimov(AT)earthlink.net), Sep 20 2006
%E a(13), a(30),... corrected.  _R. J. Mathar_, Aug 26 2020
