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 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  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 A306601 A237851 Adjacent sequences:  A127821 A127822 A127823 * A127825 A127826 A127827 KEYWORD nice,nonn,tabf,look AUTHOR T. D. Noe, Jan 31 2007 STATUS approved

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Last modified November 11 15:41 EST 2019. Contains 329017 sequences. (Running on oeis4.)