1,2

This sequence is the base sequence of the map : a(n) = a(n-1)+ [least square > a(n-1)] if a(n) is not divisible by Y, else a(n)=a(n-1)/Y , where Y is a positive integer. Experimental results shows this map converges to a periodic orbit for all Y. What is the number and length of periodic orbits for different Y ? What is the trajectory of some input under the map? If Y=2, the map converges to two periodic orbits : {1--5--14--7--16--8--4--2} and {11--27--63--127--271--560--280--140--70--35--71--152--76--38--19--44--22} which length is L1=8, L2=17. Two examples of trajectories for initial value 9 resp. 13 under the map for Y=2 : 9--25--61--125--269--558--279--568--284--142--{76--38--19--44--22--11--27--63--127--271--560--280--140--70--35--71--152} , 13--29--65--146--73--154--77--158--79--160--80--40--20--10--{5--14--7--16--8--4--2--1}.

J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.

Cf. A006370, A048761

Sequence in context: A074784 A109678 A000330 this_sequence A070129 A081861 A023652

Adjacent sequences: A166065 A166066 A166067 this_sequence A166069 A166070 A166071

nonn

Ctibor O. ZIZKA (c.zizka(AT)email.cz), Oct 06 2009