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%I #19 Apr 27 2020 17:34:24
%S 2,1212222,22,12222,21212222,1212122122212222,222,1221212122122212222,
%T 212222,12122122212222,221212222,122212222,21212122122212222,
%U 12121212222212222,2222,122122212222,21221212122122212222,12122212122122212222,2212222,1222222,212122122212222
%N "Wondrous representation" [right to left] of positive integer n, n >= 2.
%C Start with k = 1; right to left "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: 122.)
%C "Wondrous numbers" (Hofstadter, 1979, pp. 400-401) are positive integers with a Collatz trajectory that eventually reaches 1.
%C 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.
%C Reading the "digits" left to right gives the Collatz trajectory of n, n >= 2. Start with n; left to right "digits": 2 means k <= k/2, 1 means k <= 3k+1.
%C For a representation to be well-formed, we can only prepend a "digit" 1 if the number reached to the right is congruent to 4 (mod 6), yielding an odd number after prepending 1. We can prepend "digit" 2 without any restriction. Thus a(n) is odd iff it starts with 1.
%D Douglas R. Hofstadter, "Gödel, Escher, Bach: an Eternal Golden Braid." New York: Basic Books, 1979.
%H Andrew Howroyd, <a href="/A303433/b303433.txt">Table of n, a(n) for n = 2..1000</a>
%e a(3) = 1212222: [right to left] 3 <= 10 <= 5 <= 16 <= 8 <= 4 <= 2 <= (1).
%o (PARI) a(n)={my(L=List()); while(n<>1, listput(L, 2-n%2); n=if(n%2, n*3+1, n/2)); fromdigits(Vec(L))} \\ _Andrew Howroyd_, Apr 27 2020
%Y "Wondrous representation" [left to right]: A303255.
%K nonn
%O 2,1
%A _Daniel Forgues_, Apr 23 2018
%E Term a(18) and beyond from _Andrew Howroyd_, Apr 27 2020
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