A246646 Irregular triangle T(n,m) with sieved modified Collatz sequences for k = A246647(n), n >= 1, m = 1, ..., A248154(n).

2, 1, 3, 5, 8, 4, 2, 6, 3, 7, 11, 17, 26, 13, 20, 10, 5, 9, 14, 7, 12, 6, 15, 23, 35, 53, 80, 40, 20, 16, 8, 18, 9, 19, 29, 44, 22, 11, 21, 32, 16, 24, 12, 25, 38, 19, 27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 206, 103, 155, 233, 350, 175, 263, 395, 593, 890, 445, 668, 334, 167, 251, 377, 566, 283, 425, 638, 319, 479, 719, 1079, 1619, 2429, 3644, 1822, 911, 1367, 2051, 3077, 4616, 2308, 1154, 577, 866, 433, 650, 325, 488, 244, 122, 61, 92, 46, 23

Offset: 1

Author

Wolfdieter Lang, Oct 02 2014

Keywords

• nonn
• tabf
• easy

Comments

The row length sequence for this irregular triangle is A248154.

The (modified or Terras) Collatz map is T(k) = (3*k +1)/2 if k is odd and T(k) = k/2 if k is even. See the array A070168.

The present irregular array starts with row n=1 for k=2 with 2, 1 and ends because the next number would be 2 which appeared already in this row (this is the trivial cycle). Row n=2 for k=3 is then 3, 5, 8, 4, 2 and stops with 2 which is the first number in this row which appeared already in row k=1. A row for k=4 does not show up because 4 already appeared in the row for k=3. Also no row for k=5 appears. Row n=3 is for k=6 with 6,3, etc.

In this way a 'minimal' Collatz table is build. The Collatz conjecture is that every positive integer is present (the end numbers in each row n >= 2 appear exactly twice).

Examples

			The irregular triangle T(n,m) begins:
n,   k \ m
1,   2:   2   1
2,   3:   3   5  8  4  2
3,   6:   6   3
4,   7:   7  11 17 26 13 20 10  5
5,   9:   9  14  7
6,  12:  12   6
7,  15:  15  23 35 53 80 40 20
8,  16:  16   8
9,  18:  18   9
10, 19:  19  29 44 22 11
11, 21:  21  32 16
12, 24:  24  12
13, 25:  25  38 19
...
Row n=14, k=27: 27 41 62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23;
Row n=15, k=28: 28 14;
Row n=16, k=30: 30 15; ...
The complete modified Collatz iteration until 1 is reached is obtained, for example for k=19, as follows: 19  29 44 22 11, (11) 17 26 13 20 10 5, (5) 8 4 2, (2) 1, that is 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1, which is row n=19 of A070168.
		

Links

• Eric Weisstein's World of Mathematics, Collatz Problem,

Crossrefs

Cf. A246647, A248154, A070168.