OFFSET

2,1

COMMENTS

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.

REFERENCES

Douglas R. Hofstadter, "GĂ¶del, 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) = 2222121: [left to right] (1) => 2 => 4 => 8 => 16 => 5 => 10 => 3.

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(Vecrev(L))} \\ Andrew Howroyd, Apr 27 2020

CROSSREFS

KEYWORD

nonn

AUTHOR

Daniel Forgues, Apr 24 2018

EXTENSIONS

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

STATUS

approved