A083514 Number of steps for iteration of map x -> (4/3)*ceiling(x) to reach an integer > 3n+1 when started at 3n+1, or -1 if no such integer is ever reached.

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

Offset: 0

Author

Benoit Cloitre, Sep 28 2003

Keywords

• nonn
• easy

Comments

It is conjectured that an integer is always reached.

Also number of steps for iteration of map x -> (4/3)*floor(x) to reach an integer when started at 3n+4.

Links

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

Crossrefs

Equals A085068(3n+1).

Programs

Maple

b:= proc(n) local i; n; for i do 4/3*ceil(%);
      if %::integer then return i fi od
    end:
a:= n-> b(3*n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 01 2021

PARI

a(n)=if(n<0,0,c=(3*n+1)*4/3; x=1; while(frac(c)>0,c=4/3*ceil(c); x++); x)

PARI

a(n)=if(n<0,0,c=(3*n+4)*4/3; x=1; while(frac(c)>0,c=4/3*floor(c); x++); x)

Formula

a(3n+1)=2.