

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 <= (k1)/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. 400401) 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 wellformed, 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, "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) = 1212222: [right to left] 3 <= 10 <= 5 <= 16 <= 8 <= 4 <= 2 <= (1).


PROG

(PARI) a(n)={my(L=List()); while(n<>1, listput(L, 2n%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



