A028398 -id:A028398 - cp's OEIS frontend

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

J. H. Conway

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

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]

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

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.

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

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) ];

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 *)

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

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

A028395 Iterate the map in A006368 starting at 14.

Original entry on oeis.org

14, 21, 16, 24, 36, 54, 81, 61, 46, 69, 52, 78, 117, 88, 132, 198, 297, 223, 167, 125, 94, 141, 106, 159, 119, 89, 67, 50, 75, 56, 84, 126, 189, 142, 213, 160, 240, 360, 540, 810, 1215, 911, 683, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

J. H. Conway

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A028393.

Cf. A180853, A180864, A182205; A028398(5) = 14.

a028395 n = a028395_list !! n
a028395_list = iterate a006368 14  -- Reinhard Zumkeller, Apr 18 2012

SubstitutionSystem[{n_ :> If[EvenQ[n], 3n/2, Round[3n/4]]}, {14}, 60] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

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

A180853 Trajectory of 4 under map n->A006368(n).

Original entry on oeis.org

4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5, 4, 6, 9, 7, 5

Offset: 0

N. J. A. Sloane, Jan 22 2011

The trajectory of 8 is a famous unsolved problem - see A028393.

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.

J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1)

Cf. A028397; A028398(3) = 4.

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

a180853 n = a180853_list !! n
a180853_list = iterate a006368 4  -- Reinhard Zumkeller, Apr 18 2012

Table[{4, 6, 9, 7, 5}, {21}] // Flatten   (* Jean-François Alcover, Jun 10 2013 *)
PadRight[{},120,{4,6,9,7,5}] (* Harvey P. Dale, Jul 11 2020 *)

Vec((-4-6*x-9*x^2-7*x^3-5*x^4)/((x-1)*(1+x+x^2+x^3+x^4))+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

Periodic with period of length 5.
G.f.: ( -4-6*x-9*x^2-7*x^3-5*x^4 ) / ( (x-1)*(1+x+x^2+x^3+x^4) ). - R. J. Mathar, Mar 10 2011
a(n+1) = A006368(a(n)).
a(n) = a(n-5). - Wesley Ivan Hurt, Apr 26 2021

A182205 Iterate the map in A006368 starting at 40.

Original entry on oeis.org

40, 60, 90, 135, 101, 76, 114, 171, 128, 192, 288, 432, 648, 972, 1458, 2187, 1640, 2460, 3690, 5535, 4151, 3113, 2335, 1751, 1313, 985, 739, 554, 831, 623, 467, 350, 525, 394, 591, 443, 332, 498, 747, 560, 840, 1260, 1890, 2835, 2126, 3189, 2392, 3588, 5382

Offset: 0

Reinhard Zumkeller, Apr 18 2012

nonn

Like for iterations with starting points 8 or 14, it is conjectured that also this trajectory never repeats.

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.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.

Cf. A028393, A028395, A180853, A180864, A182205; A028398(6) = 40.

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

a182205 n = a182205_list !! n
a182205_list = iterate a006368 40

SubstitutionSystem[{n_ :> If[EvenQ[n], 3 n/2, Round[3 n/4]]}, {40}, 60] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

a(n+1) = A006368(a(n)), with a(0) = 40.

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A028393 Iterate the map in A006368 starting at 8.

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A028395 Iterate the map in A006368 starting at 14.

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A180853 Trajectory of 4 under map n->A006368(n).

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A182205 Iterate the map in A006368 starting at 40.

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