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A127824
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Triangle in which row n is a sorted list of all numbers having total stopping time n in the Collatz (or 3x+1) iteration.
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17
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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)
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OFFSET
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0,2
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COMMENTS
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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.
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
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REFERENCES
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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MATHEMATICA
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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}]]]
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PROG
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(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]
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CROSSREFS
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Cf. A006577 (total stopping time of n), A088975 (traversal of the Collatz tree).
Last elements of rows give A000079.
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KEYWORD
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nice,nonn,tabf,look,changed
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AUTHOR
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STATUS
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approved
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