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

0,2

Comments

It is conjectured that an integer is always reached.

Links

Table of n, a(n) for n=0..100.

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

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 *)

Prog

(Python3)
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

Crossrefs

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

Sequence in context: A349930 A274512 A175947 * A265027 A079587 A237881

Adjacent sequences: A085065 A085066 A085067 * A085069 A085070 A085071

Keyword

nonn

Author

N. J. A. Sloane, Aug 11 2003

Status

approved