OEIS/A307407

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Name

Irregular table of rows with terms which map uniquely to the nodes in the graph of the "3x+1" (or Collatz) problem.

Data

16, 4, 5, 1, 10, 2, 3, 40, 12, 13, 64, 20, 21, 88, 28, 29, 9, 58, 112, 36, 37, 136, 44, 45, 160, 52, 53, 17, 106, 34, 35, 11, 70, 22, 23, 7, 46, 14, 15, 184, 60, 61

Offset 1,1

Comments

The sequence is the flattened form of an irregular table S(i, j) (see the example below) which has rows i >= 1 consisting of subsequences of varying length. We define three mappings:

d (for "divide") maps n -> (n - 1)/3 if n == 1 mod 3, 
m (for "multiply") maps n = n/2 for all n,
s (for "squeeze") is dm and maps n ->  ((n - 1) / 3) * 2. 

We note the mappings as infix operators, for example 16 d 5 and 5 m 10. For any positive row number i (also the segment "index" or "left side") as starting value, we compute the elements S[i, j] by the mappings shown in the column "Map" of the following table:

j | Map  | Form of i    |  S(i, j)     |  Residues  | Residues not yet covered
--+------+ -------------+--------------+------------+-------------------------
1 |      |  1 * k +  1  |   4 * k +  3 |   3 mod  4 |   0,  1,  2     mod  4
2 | / 3  |  3 * k +  1  |   4 * k +  1 |   1 mod  4 |   0,  2,  4,  6 mod  8
3 | * 2  |  3 * k +  1  |   8 * k +  2 |   2 mod  8 |   0,  4,  6     mod  8
4 | / 3  |  9 * k +  7  |   8 * k +  6 |   6 mod  8 |   0,  4,  8, 12 mod 16
5 | * 2  |  9 * k +  7  |  16 * k + 12 |  12 mod 16 |   0,  4,  8     mod 16
6 | / 3  | 27 * k +  7  |  16 * k +  4 |   4 mod 16 |   0,  8, 16, 24 mod 32
7 | * 2  | 27 * k +  7  |  32 * k +  8 |   8 mod 32 |   0, 16, 24     mod 32
8 | / 3  | 81 * k + 61  |  32 * k + 24 |  24 mod 32 |   0, 16, 32, 48 mod 64
9 | * 2  | 81 * k + 61  |  64 * k + 48 |  48 mod 64 |   0, 16, 32     mod 64
..| ...  |  e * k +  f  |   g * k +  m |   m mod  g |   0, ...

the following series of mappings:

j = 1: mm, j 2 as defined by the following algorithm:

Construction and proof are similiar to that in A322469, which is more simple.

Example

Table S(i, j) begins:

 i\j    1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
 ----------------------------------------------------------------
 1:    16   4   5   1  10   2   3
 2:    40  12  13
 3:    64  20  21
 4:    88  28  29   9  58
 5:   112  36  37
 6:   136  44  45
 7:   160  52  53  17 106  34  35  11  70  22  23   7  46  14  15
 8:   184  60  61

Crossrefs