A127824 Triangle in which row n is a sorted list of all numbers having total stopping time n in the Collatz (or 3x+1) iteration.

1, 2, 4, 8, 16, 5, 32, 10, 64, 3, 20, 21, 128, 6, 40, 42, 256, 12, 13, 80, 84, 85, 512, 24, 26, 160, 168, 170, 1024, 48, 52, 53, 320, 336, 340, 341, 2048, 17, 96, 104, 106, 113, 640, 672, 680, 682, 4096, 34, 35, 192, 208, 212, 213, 226, 227, 1280, 1344, 1360, 1364

Offset

0,2

Comments

The length of each row is A005186(n). The largest number in row n is 2^n. The second-largest number in row n is A000975(n-2) for n>4. The smallest number in row n is A033491(n). The Collatz conjecture asserts that every positive integer occurs in some row of this triangle.

n is an element of row number A006577(n). - Reinhard Zumkeller, Oct 03 2012

Conjecture: The numbers T(n, 1),...,T(n, k_n) of row n are arranged in non-overlapping clusters of numbers which have the same order of magnitude and whose Collatz trajectories to 1 have the same numbers of ups and downs. The highest cluster of row n is just the number 2^n, the trajectory to 1 of which has n-1 downs and no ups. The second highest cluster of row n consists of the numbers T(n, k_n - r) = 4^(r - 1) * t(n - 2*r + 2) for 1 <= r <= (n - 3) / 2, where t(k) = (2^k - (-1)^k - 3) / 6. These have n-2 downs and one up. The largest and second largest number of this latter cluster are given by A000975 and A153772. - Markus Sigg, Sep 25 2020

References

See also A006577.

Links

Alois P. Heinz, Rows n = 0..42, flattened (first 31 rows from T. D. Noe)

Paul Andaloro, On total stopping times under 3x+1 iteration, Fib. Quar. 38 (1) (2000) 73.

Jason Davies, Collatz Graph: All Numbers Lead to One

Wolfdieter Lang, On Collatz Words, Sequences, and Trees, Journal of Integer Sequences, Vol 17 (2014), Article 14.11.7.

Markus Sigg, On the cluster structures in Collatz preimages, arXiv:2012.07839 [math.GM], 2020.

Formula

Suppose S is the list of numbers in row n. Then the list of numbers in row n+1 is the union of each number in S multiplied by 2 and the numbers (x-1)/3, where x is in S, with x=1 (mod 3) and where (x-1)/3 is an odd number greater than 1.

Example

The triangle starts:
   0:   1
   1:   2
   2:   4
   3:   8
   4:  16
   5:   5   32
   6:  10   64
   7:   3   20   21  128
   8:   6   40   42  256
   9:  12   13   80   84   85  512
  10:  24   26  160  168  170 1024
  11:  48   52   53  320  336  340  341 2048
  12:  17   96  104  106  113  640  672  680  682 4096
- Reinhard Zumkeller, Oct 03 2012

Mathematica

s={1}; t=Flatten[Join[s, Table[s=Union[2s, (Select[s, Mod[ #, 3]==1 && OddQ[(#-1)/3] && (#-1)/3>1&]-1)/3]; s, {n, 13}]]]

Prog

(Haskell)
import Data.List (union, sort)
a127824 n k = a127824_tabf !! n !! k
a127824_row n = a127824_tabf !! n
a127824_tabf = iterate f [1] where
   f row = sort $ map (* 2) row `union`
                  [x' | x <- row, let x' = (x - 1) `div` 3,
                        x' * 3 == x - 1, odd x', x' > 1]
-- Reinhard Zumkeller, Oct 03 2012

Crossrefs

Cf. A006577 (total stopping time of n), A088975 (traversal of the Collatz tree).

Column k=1 gives A033491.

Last elements of rows give A000079.

Row lengths give A005186.

Row sums give A337673(n+1).

Sequence in context: A225570 A352391 A178170 * A088975 A306601 A237851

Adjacent sequences: A127821 A127822 A127823 * A127825 A127826 A127827

Keyword

nice,nonn,tabf,look

Author

T. D. Noe, Jan 31 2007

Status

approved