A074078 Number of steps to reach an integer starting with s = n/3 and iterating the map x -> s*ceiling(x).

0, 2, 4, 0, 1, 1, 0, 13, 2, 0, 3, 2, 0, 1, 1, 0, 2, 4, 0, 8, 5, 0, 1, 1, 0, 7, 9, 0, 2, 7, 0, 1, 1, 0, 3, 2, 0, 6, 2, 0, 1, 1, 0, 2, 3, 0, 10, 3, 0, 1, 1, 0, 3, 3, 0, 2, 3, 0, 1, 1, 0, 5, 2, 0, 5, 2, 0, 1, 1, 0, 2, 10, 0, 3, 7, 0, 1, 1, 0, 8, 4, 0, 2, 6, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 1, 1, 0, 2, 5, 0, 4, 6, 0, 1, 1, 0

Offset: 3

Author

N. J. A. Sloane, Sep 16 2002

Keywords

• nonn

Examples

			s = 5/3 -> 10/3 -> 20/3 -> 35/3 -> 20, so a(5) = 4.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 3..10002

Crossrefs

Cf. A072340, A074090, A074091, A074096. Records are in A074097, A074098.

First integer reached: A081852.

Programs

Maple

f := proc(b1,b2) local c1,c2,t1,t2,t3,t4,i; c1 := numer(b1/b2); c2 := denom(b1/b2); i := 0; while c2 <> 1 do i := i+1; t1 := ceil(c1/c2); t2 := b1*t1; t3 := t2/b2; c1 := numer(t3); c2 := denom(t3); od: RETURN(i); end; [seq(f(n,3),n=4..120)];

Mathematica

ce[n_] := Length[NestWhileList[(n/3)*Ceiling[#] &, n/3, ! IntegerQ[#] &]] - 1; Table[ce[n], {n, 3, 108}] (* Jayanta Basu, Jul 30 2013 *)

Python

from math import ceil
from fractions import Fraction
def a(n):
    s = Fraction(n, 3)
    x, c = s, 0
    while x.denominator != 1:
        U = ceil(x)
        x, c = U*s, c+1
    return c
print([a(n) for n in range(3, 109)]) # Michael S. Branicky, Jan 09 2025