A085068 Number of steps >= 1 for iteration of map x -> (4/3)*ceiling(x) to reach an integer when started at n, or -1 if no such integer is ever reached.

1, 3, 2, 1, 2, 9, 1, 8, 3, 1, 7, 2, 1, 2, 6, 1, 3, 4, 1, 5, 2, 1, 2, 3, 1, 6, 4, 1, 3, 2, 1, 2, 4, 1, 5, 3, 1, 4, 2, 1, 2, 4, 1, 3, 8, 1, 4, 2, 1, 2, 3, 1, 4, 7, 1, 3, 2, 1, 2, 7, 1, 4, 3, 1, 9, 2, 1, 2, 6, 1, 3, 6, 1, 5, 2, 1, 2, 3, 1, 6, 5, 1, 3, 2, 1, 2, 8, 1, 5, 3, 1, 5, 2, 1, 2, 5, 1, 3, 4, 1, 6

Offset: 0

Author

N. J. A. Sloane, Aug 11 2003

Keywords

• nonn
• changed

Comments

It is conjectured that an integer is always reached.

Links

• J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A085058, A085071, A085328, A085330, A083514.

Programs

Maple

f := x->(4/3)*ceil(x); g := proc(n) local t1,c; global f; t1 := f(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := f(t1); od; RETURN([c,t1]); end;
# second Maple program:
a:= proc(n) local i; n; for i do 4/3*ceil(%);
      if %::integer then return i fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 01 2021

Mathematica

f[n_] := Block[{c = 1, k = 4 n/3}, While[ ! IntegerQ@k, c++; k = 4 Ceiling@k/3]; c]; Table[f@n, {n, 0, 104}] (* Robert G. Wilson v *)

Python

from fractions import Fraction
def A085068(n):
    c, x, m = 1, Fraction(4*n,3), Fraction(4,3)
    while x.denominator > 1:
        x = m*x._ceil_()
        c += 1
    return c # Chai Wah Wu, Mar 01 2021