Although the forward equation is quite easy to calculate, calculating the
precursors of a given number is more interesting. Looking at "1", we see that it
is preceded by the infinite sequence of powers of 2. We are focusing on odd
predecessors, so we have to find powers of two that fit the equation
2k = 3x+1
=> x = (2k-1) / 3.
Where x is an integer. Only even values of k lead to integer solutions, giving;
| k
|
(2k-1) / 3
|
| 2
|
(4-1)/3 = 1
|
| 4
|
(16-1)/3 = 5
|
| 6
|
(64-1)/3 = 21
|
| 8
|
(256-1)/3 = 85
|
|
|
| k
|
(2k-1) / 3
|
| 10 |
(1024-1)/3 = 341 |
| 12 |
(4096-1)/3 = 1,365 |
| 14 |
(16384-1)/3 = 5,461 |
| 16 |
(65536-1)/3 = 21,845 |
|
Etc. For
numbers other than 1, say 'A', we get the similar equation
A.2k = 3x+1, or x = (A.2k-1) / 3.
Calculating x for a few
values of A and k we can see a pattern;
| x(A,k)
|
k=2
|
k=4
|
k=6
|
| A=1
|
1
|
5
|
21
|
| A=7
|
9
|
37
|
149
|
| A=13 |
17 |
69 |
277 |
|
| x(A,k)
|
k=1
|
k=3
|
k=5
|
| A=5
|
3
|
13
|
53
|
| A=11
|
7
|
29
|
117
|
| A=17
|
11
|
45
|
181
|
|
When A is a multiple of three we get no predecessors, as a multiple of three
can't be of the form 3x+1. When A is one more than a multiple of 6, k will be
even, and when it is one less than a multiple of 6, k will be odd. The smallest
precursors are therefore: 6n+1 -> 8n+1; 6n+5 -> 4n +
3.
When reading down the columns the values increase linearly in steps of 2(k+1),
whereas across the table they increase geometrically as x -> 4x+1.
The next two tables are extensions of the previous tables, but now the
numbers are colour-coded according to their remainders when divided by 6.
| A\k |
|
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
| 1 |
|
1 |
5 |
21 |
85 |
341 |
1,365 |
5,461 |
21,845 |
87,381 |
| 7 |
|
9 |
37 |
149 |
597 |
2,389 |
9,557 |
38,229 |
152,917 |
611,669 |
| 13 |
|
17 |
69 |
277 |
1,109 |
4,437 |
17,749 |
70,997 |
283,989 |
1,135,957 |
| 19 |
|
25 |
101 |
405 |
1,621 |
6,485 |
25,941 |
103,765 |
415,061 |
1,660,245 |
| 25 |
|
33 |
133 |
533 |
2,133 |
8,533 |
34,133 |
136,533 |
546,133 |
2,184,533 |
| 31 |
|
41 |
165 |
661 |
2,645 |
10,581 |
42,325 |
169,301 |
677,205 |
2,708,821 |
| A\k
|
|
1
|
3
|
5
|
7
|
9
|
11
|
13 |
15 |
17 |
| 5 |
|
3 |
13 |
53 |
213 |
853 |
3,413 |
13,653 |
54,613 |
21,8453 |
| 11 |
|
7 |
29 |
117 |
469 |
1,877 |
7,509 |
30,037 |
120,149 |
480,597 |
| 17 |
|
11 |
45 |
181 |
725 |
2,901 |
11,605 |
46,421 |
185,685 |
742,741 |
| 23 |
|
15 |
61 |
245 |
981 |
3,925 |
15,701 |
62,805 |
251,221 |
1,004,885 |
| 29 |
|
19 |
77 |
309 |
1,237 |
4,949 |
19,797 |
79,189 |
316,757 |
1,267,029 |
Notice that numbers of the form 6n+5 have a smaller odd predecessor. This can
be used to generate arbitrary long sequences of
numbers where (3x+1)/2 is odd and (3*(3x+1)/2+1) is larger than 3x+1.
The colour patterns
repeat every 3 rows, as we can see from the following tables
| A\k
|
|
2
|
4
|
6
|
| 18n+1 |
|
24n+1 = 6.(4n)+1 |
96n+5 = 6.(16n)+5 |
384n+21 = 6.(64n+3)+3 |
| 18n+7 |
|
24n+9 = 6.(4n+1)+3 |
96n+37 = 6.(16n+6)+1 |
384n+149 = 6.(64n+24)+5 |
| 18n+13 |
|
24n+17 = 6.(4n+2)+5 |
96n+69 = 6.(16n+11)+3 |
384n+277 = 6.(64n+46)+1 |
| A\k
|
|
1
|
3
|
5
|
| 18n+5 |
|
12n+3 = 6.(2n)+3 |
48n+13 = 6.(8n+2)+1 |
192n+53 = 6.(32n+8)+5 |
| 18n+11 |
|
12n+7 = 6.(2n+1)+1 |
48n+29 = 6.(8n+4)+5 |
192n+117 = 6.(128n+19)+3 |
| 18n+17 |
|
12n+11 = 6.(2n+1)+5 |
48n+45 = 6.(8n+7)+3 |
192n+181 = 6.(128n+30)+1 |
