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A303433 "Wondrous representation" [right to left] of positive integer n, n >= 2. 2
2, 1212222, 22, 12222, 21212222, 1212122122212222, 222, 1221212122122212222, 212222, 12122122212222, 221212222, 122212222, 21212122122212222, 12121212222212222, 2222, 122122212222, 21221212122122212222, 12122212122122212222, 2212222, 1222222, 212122122212222 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.)

"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" 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.

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.

REFERENCES

Douglas R. Hofstadter, "Gödel, Escher, Bach: an Eternal Golden Braid." New York: Basic Books, 1979.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 2..1000

EXAMPLE

a(3) = 1212222: [right to left] 3 <= 10 <= 5 <= 16 <= 8 <= 4 <= 2 <= (1).

PROG

(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

CROSSREFS

"Wondrous representation" [left to right]: A303255.

Sequence in context: A324439 A218169 A168535 * A253264 A124368 A272238

Adjacent sequences:  A303430 A303431 A303432 * A303434 A303435 A303436

KEYWORD

nonn

AUTHOR

Daniel Forgues, Apr 23 2018

EXTENSIONS

Term a(18) and beyond from Andrew Howroyd, Apr 27 2020

STATUS

approved

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Last modified May 28 12:31 EDT 2020. Contains 334681 sequences. (Running on oeis4.)