login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 10
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 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

See also A006577.

LINKS

T. D. Noe, Rows n=0..30 of triangle, flattened

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.

FORMULA

Suppose S is the list of numbers in row n, then the list of numbers in row n+1 is the union of (a) each number in S multiplied by 2 and (b) numbers (x-1)/3 where x is in S, with x=1 (mod 3) and (x-1)/3 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).

Sequence in context: A269305 A225570 A178170 * A088975 A237851 A167425

Adjacent sequences:  A127821 A127822 A127823 * A127825 A127826 A127827

KEYWORD

nice,nonn,tabf,look

AUTHOR

T. D. Noe, Jan 31 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 23 15:54 EDT 2017. Contains 292361 sequences.