Friday, 28 April 2017

Collatz Conjecture Revisited (part 2): 3n+3

I've previously looked at the Collatz conjecture, (3n+1) and I have revisited it before, too (5n+1).  Now, I would like to revisit it again.

The Collatz Conjecture states that when you take a number, and if it's even then divide by two, or if it's odd then multiply by three and add one, then you will eventually reach 1.  There's no proof (yet), but it holds for all numbers that have been tested.



I extended this in a previous post, and looked at the case of multiplying by five (instead of three) and adding one, and identified two loops and a growing series.

In this post, I will share my findings on another alternative, which is "3n+3".  [3n+2 doesn't work, since if n is odd, then 3n + 2 will also be odd].


3n+3


3n+3 has one loops which covers all numbers I have tested.

The simple loop/termination is [1], 6, 3, 12, 6, 3, 12.

There are various ways into this loop, in particular,

10, 5, 18, 9, 30, 15, 48, 24, 12, 6, 3, 12 etc.
7, 30, 15, 48 etc.
11, 36, 18, 9, 30, 15, etc.
13, 42, 21, 66, 33, 102, 51, 156, 78, 39, 120, 60, 30, 15, etc.

Interestingly, many of the starting numbers reach a common maximum value of 27696 before coming back down to 1.    This is first seen for an initial n=53.

For larger values of n, there is a longer sequence.  The graph below shows the maximum value of n (vertical axis) for different start values of n (between 101 and 241, as example material).  Note how 27696 predominates as the largest value reached.


The sequence from 27696 is:

27696, 13848, 6924, 3462, 1731, 5196, 2598, 1299, 3900, 1950, 975, 2928, 1464, 732, 366, 183, 552, 276, 138, 69, 210, 105, 318, 159, 480, 240, 120, 60, 30, 15, 48, 24, 12, 6, 3

27696 is seen for the following initial values of n:

53, 61, 81, 93, 107, 109, 123, 125, 141, 145, 163, 165, 181, 187, 189 (and others).

For values above 27696
I have not explored extensively above 27696, but there is a cluster of initial values that have the same new peak.  The cluster is around 27754:  27754, 27755 and 27757 all have the same maximum, which is 2026128.  The highest peak I have observed so far is for 27729, which reaches a height of 2698752.

To close, the full sequence for 27729 is:

27729, 83190, 41595, 124788, 62394, 31197, 93594, 46797, 140394, 70197, 210594, 105297, 315894, 157947, 473844, 236922, 118461, 355386, 177693, 533082, 266541, 799626, 399813, 1199442, 599721, 1799166, 899583, 2698752, 1349376, 674688, 337344, 168672, 84336, 42168, 21084, 10542, 5271, 15816, 7908, 3954, 1977, 5934, 2967, 8904, 4452, 2226, 1113, 3342, 1671, 5016, 2508, 1254, 627, 1884, 942, 471, 1416, 708, 354, 177, 534, 267, 804, 402, 201, 606, 303, 912, 456, 228, 114, 57, 174, 87, 264, 132, 66, 33, 102, 51, 156, 78, 39, 120, 60, 30, 15, 48, 24, 12, 6, 3

I shall continue to explore 3n+3, and also too compare the data and sequences with 3n+5.