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A303255
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"Wondrous representation" [left to right] of positive integer n, n >= 2.
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2
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2, 2222121, 22, 22221, 22221212, 2222122212212121, 222, 2222122212212121221, 222212, 22221222122121, 222212122, 222212221, 22221222122121212, 22221222221212121, 2222, 222212221221, 22221222122121212212, 22221222122121222121, 2222122, 2222221, 222212221221212
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OFFSET
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2,1
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COMMENTS
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Start with k = 1; left to right "digits": 2 means k <= 2k, 1 means k <= (k-1)/3. (1 has the empty "wondrous representation," since it is "wondrous" by definition ... although, for a nonempty representation, we could [in a kludgy way] represent 1 using the trivial cycle: 221.)
"Wondrous numbers" (Hofstadter, 1979, pp. 400-401) are positive integers with a Collatz trajectory that eventually reaches 1.
According to the Collatz conjecture, every positive integer is "wondrous" (none is "unwondrous"). Thus, every positive integer n >= 2 is conjectured to have a "wondrous representation," which is then unique.
Reading the "digits" right to left gives the Collatz trajectory of n, n >= 2. Start with n; right to left "digits": 2 means k <= k/2, 1 means k <= 3k+1.
For a representation to be well-formed, we can only append a "digit" 1 if the number reached to the left is congruent to 4 (mod 6), yielding an odd number after appending 1. We can append "digit" 2 without any restriction. Thus a(n) is odd iff it ends with 1.
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REFERENCES
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Douglas R. Hofstadter, "Gödel, Escher, Bach: an Eternal Golden Braid." New York: Basic Books, 1979.
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LINKS
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EXAMPLE
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a(3) = 2222121: [left to right] (1) => 2 => 4 => 8 => 16 => 5 => 10 => 3.
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PROG
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(PARI) a(n)={my(L=List()); while(n<>1, listput(L, 2-n%2); n=if(n%2, n*3+1, n/2)); fromdigits(Vecrev(L))} \\ Andrew Howroyd, Apr 27 2020
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CROSSREFS
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"Wondrous representation" [right to left]: A303433.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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