A028393 Iterate the map in A006368 starting at 8.

8, 12, 18, 27, 20, 30, 45, 34, 51, 38, 57, 43, 32, 48, 72, 108, 162, 243, 182, 273, 205, 154, 231, 173, 130, 195, 146, 219, 164, 246, 369, 277, 208, 312, 468, 702, 1053, 790, 1185, 889, 667, 500, 750, 1125, 844, 1266, 1899, 1424, 2136, 3204, 4806, 7209, 5407

Offset: 0

Author

J. H. Conway

Keywords

• nonn
• look

Comments

It is conjectured that this trajectory never repeats, but no proof of this has been found. - N. J. A. Sloane, Jul 14 2009

References

• J. H. Conway, Unpredictable iterations, in Proc. Number Theory Conf., Boulder, CO, 1972, pp. 49-52. - N. J. A. Sloane, Oct 04 2012
• R. K. Guy, Unsolved Problems in Number Theory, E17. - N. J. A. Sloane, Oct 04 2012
• J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 5. [From N. J. A. Sloane, Jan 21 2011]

Links

• Markus Sigg, Table of n, a(n) for n = 0..9999 (terms 0..1000 from T. D. Noe)
• J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.
• D. Gale, Tracking the Automatic Ant And Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 16. [From _N. J. A. Sloane_, Jul 14 2009]
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006368, A028394, A180864.

Cf. A028395, A180853, A182205; A028398(4) = 8.

Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

Programs

Haskell

a028393 n = a028393_list !! n
a028393_list = iterate a006368 8  -- Reinhard Zumkeller, Apr 18 2012

Maple

F := proc(n) option remember; if n = 0 then 8 elif 3*F(n-1) mod 2 = 0 then 3*F(n-1)/2 else round(3*F(n-1)/4); fi; end; [ seq(F(i),i=0..80) ];

Mathematica

f[n_?EvenQ] := 3*n/2; f[n_] := Round[3*n/4]; a[0] = 8; a[n_] := a[n] = f[a[n - 1]]; Table[a[n], {n, 0, 52}]  (* Jean-François Alcover, Jun 10 2013 *)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def F(n):
    if n == 0: return 8
    elif 3*F(n-1)%2 == 0: return 3*F(n-1)//2
    else: return (3*F(n-1)+1)//4
print([F(i) for i in range(81)]) # Michael S. Branicky, Aug 12 2021 after J. H. Conway

Formula

a(n+1) = A006368(a(n)).