A129121 Starting numbers x for which the trajectory of x->R(A006370(x)) ends in the cycle 1->4->2->1.

1, 2, 3, 4, 6, 8, 12, 13, 16, 17, 20, 25, 33, 34, 36, 37, 39, 40, 42, 48, 50, 52, 59, 60, 62, 66, 67, 69, 71, 76, 80, 81, 83, 86, 89, 95, 99, 101, 103, 104, 107, 109, 117, 118, 120, 121, 122, 125, 126, 132, 133, 134, 135, 136, 137, 139, 142, 143, 145, 149, 152, 157

Offset

1,2

Comments

Define the map x-> A004086(A006370(x)) = E(x), which is the 3x+1 "Collatz" operation followed by digit reversal.

Examples of trajectories starting from small integers are:

1->4->2->1->4->2->1->4->2->1->4->

2->1->4->2->1->4->2->1->4->2->1->

3->1->4->2->1->4->2->1->4->2->1->

4->2->1->4->2->1->4->2->1->4->2->

5->61->481->4441->42331->499621->4688941->42866041->421895821->...

6->3->1->4->2->1->4->2->1->4->2->

7->22->11->43->31->49->841->4252->6212->6013->4081->44221->466231->4968931->..

8->4->2->1->4->2->1->4->2->1->4->

9->82->14->7->22->11->43->31->49->841->4252->->6212->6013->4081->44221->

10->5->61->481->4441->42331->499621->4688941->42866041->...

11->43->31->49->841->4252->6212->6013->4081->44221->466231->

12->6->3->1->4->2->1->4->2->1->4->

13->4->2->1->4->2->1->4->2->1->4->

14->7->22->11->43->31->49->841->4252->6212->6013->->4081->44221->466231->

15->64->23->7->22->11->43->31->49->841->4252->

16->8->4->2->1->4->2->1->4->2->1->

17->25->67->202->101->403->121->463->931->4972->6842->->1243->373->211->436->

18->9->82->14->7->22->11->43->31->49->841->

19->85->652->623->781->4432->6122->1603->184->29->88->

20->1->4->2->1->4->2->1->4->2->1->

When the trajectory of E(n) doesn't reach 1, it

(i) either reaches 238 and from there enters the 18 steps cycle 911, 4372, 6812, 6043, 3181, 4459, 87331, 449162, 185942, 17929, 88375, 602662, 133103, 8914, 7544, 2773 or

(ii) 247 and from there keeps indefinitely this value, or

(iii) it reaches an integer with 4 the most significant digit and 1 the least significant digit and from there keeps this structure and grows indefinitely; there is no other pair of digits with this peculiarity.

There is also a 7-cycle: 320281 -> 448069 -> 8024431 -> 49237042 -> 12581642 -> 1280926 -> 364046 -> 320281. - Charlie Neder, Jan 12 2019

Links

Table of n, a(n) for n=1..62.

Example

As commented above, starting from 1 to 4 enters the 4-cycle, so the first 4 integers are added to the sequence. Starting from 5, the trajectory grows indefinitely (does not enter the 4-cycle), so 5 is not added to the sequence.

Crossrefs

Cf. A004086, A006370, A006577.

Sequence in context: A323383 A329905 A085113 * A018556 A018387 A336503

Adjacent sequences: A129118 A129119 A129120 * A129122 A129123 A129124

Keyword

easy,base,nonn

Author

Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), May 25 2007

Extensions

Edited by R. J. Mathar, Oct 02 2009

Status

approved