A087704 Number of steps for iteration of map x -> (5/3)*floor(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

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

Offset: 2

Author

N. J. A. Sloane, Sep 29 2003

Keywords

• nonn

Comments

It is conjectured that an integer is always reached.

Links

• Robert Israel, Table of n, a(n) for n = 2..10000
• J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A087705, A087706, A087707.

Programs

Maple

f2 := proc(x,y) x*floor(y); end; r := 5/3; h := proc(x) local n,y; global r; y := f2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := f2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(h(n)[2],n=2..60)];

Python

from fractions import Fraction
def A087704(n):
    x, c = Fraction(n,1), 0
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._floor_(),3)
        c += 1
    return c # Chai Wah Wu, Sep 01 2023

Formula

a(n) = a(n + m) if a(n) > 0 and m is a (positive or negative) multiple of 3^a(n). - Robert Israel, Sep 01 2023