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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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5
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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; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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REFERENCES
| See A006577.
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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
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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.
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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}]]]
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CROSSREFS
| Cf. A006577 (total stopping time of n), A088975 (traversal of the Collatz tree).
Sequence in context: A050076 A070337 A178170 * A088975 A167425 A016018
Adjacent sequences: A127821 A127822 A127823 * A127825 A127826 A127827
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KEYWORD
| nice,nonn,tabl
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jan 31 2007
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