A087707 Number of steps for iteration of map x -> (5/3)*ceiling(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

5, 4, 1, 3, 2, 1, 2, 3, 1, 10, 4, 1, 6, 2, 1, 2, 9, 1, 3, 3, 1, 5, 2, 1, 2, 5, 1, 4, 8, 1, 3, 2, 1, 2, 3, 1, 4, 12, 1, 5, 2, 1, 2, 4, 1, 3, 3, 1, 7, 2, 1, 2, 4, 1, 5, 6, 1, 3, 2, 1, 2, 3, 1, 11, 5, 1, 4, 2, 1, 2, 6, 1, 3, 3, 1, 4, 2, 1, 2, 5, 1, 6, 4, 1, 3, 2, 1, 2, 3, 1, 6, 4, 1, 5, 2, 1, 2, 5, 1, 3

Offset: 1

Author

N. J. A. Sloane, Sep 29 2003

Keywords

• nonn

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. A087704, A087705, A087706, A087708, A087709.

Programs

Maple

c2 := proc(x,y) x*ceil(y); end; r := 5/3; ch := proc(x) local n,y; global r; y := c2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := c2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(ch(n)[2],n=1..100)];

Python

from fractions import Fraction
def A087707(n):
    x, c = Fraction(n), 0
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._ceil_(),3)
        c += 1
    return c # Chai Wah Wu, Sep 02 2023