A350651 a(n) is the size of the orbit of n under repeated application of A350229 (the sum of a number and its balanced ternary digits).

1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 7, 1, 7, 6, 6, 5, 4, 4, 4, 3, 5, 3, 4, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 6, 1

Offset: 0

Author

Rémy Sigrist, Jan 09 2022

Keywords

• nonn
• base

Comments

This sequence is well defined:

- for any w such that 3^w > 2*w + 2:

- let M(w) (resp. m(w)) be the number whose balanced ternary expansion

starts with "1T"

followed by w T's

followed by w 1's (resp. w T's),

- for any u in the interval m(w)..M(w), A350229(u) <= u <= M(w),

- also for any v < m(k), A350229(v) < m(k) + 2*w + 2 <= M(w),

- and the orbit of any number <= M(w) will be trapped in the interval 0..M(w),

- hence for any number n, we can find an appropriate w, so the orbit of n is bounded and eventually periodic (with a finite size), QED.

This sequence is unbounded.

Examples

			For n = 9:
- the orbit of 9 contains the following values:
  k  v   bter(v)  ds(v)
  -  --  -------  -----
  0   9      100      1
  1  10      101      2
  2  12      110      2
  3  14     1TTT     -2
  4  12      110      2
- so a(9) = #{ 9, 10, 12, 14 } = 4.
		

Crossrefs

Cf. A065363, A350229.

Programs

Mathematica

f[n_] := n + Total[If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[n, 3], 0] //. {x___, y_, k_ /; k > 1, z___} :> {x, y + 1, k - 3, z}]]; Array[-1 + Length@ NestWhileList[f, #, UnsameQ, All] &, 105, 0] (* Michael De Vlieger, Jan 15 2022 *)

PARI

b(n) = my (v=n, d); while (n, n=(n-d=[0,1,-1][1+n%3])/3; v+=d); v
a(n) = my (s=[]); while (!setsearch(s, n), s=setunion(s, [n]); n=b(n)); #s

Formula

a(n) = #{ A350229^k(n), k >= 0 } (where f^k corresponds to the k-th iterate of f).