A006368 -id:A006368 - 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.

A180864 Trajectory of 13 under map n->A006368(n).

Original entry on oeis.org

13, 10, 15, 11, 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, 4055, 3041, 2281, 1711, 1283, 962

Offset: 0

N. J. A. Sloane, Jan 22 2011

nonn

Merges with the trajectory of 8 after four steps - see A028393.

It is a famous unsolved problem to show that this trajectory is unbounded.

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

Cf. A006368, A028393, A028395, A028397.

Cf. A180853, A182205.

a180864 n = a180864_list !! n
a180864_list = iterate a006368 13  -- Reinhard Zumkeller, Apr 18 2012

b[n_] := If[EvenQ[n], 3n/2, Floor[(3n+2)/4]];
a[0] = 13; a[n_] := a[n] = b[a[n-1]];
Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Aug 01 2018 *)
SubstitutionSystem[{n_ :> If[EvenQ[n], 3n/2, Round[3n/4]]}, {13}, 62] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

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

A217218 Trajectory of 44 under the map k -> A006368(k).

Original entry on oeis.org

44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59, 44, 66, 99, 74, 111, 83, 62, 93, 70, 105, 79, 59

Offset: 1

N. J. A. Sloane, Oct 04 2012

Periodic with period length 12.

It is believed that this is the longest trajectory that cycles (the others are {1}, {2,3}, {4,6,9,7,5}).

See also references and links in A006368.

Colin Barker, Table of n, a(n) for n = 1..1000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.
John L Simons, Cycles and divergent trajectories for a class of permutation sequences, arXiv:2205.10582 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A006368.

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

a217218 n = a217218_list !! (n-1)
a217218_list = iterate a006368 44  -- Reinhard Zumkeller, Apr 06 2013

&cat[ [44,66,99,74,111,83,62,93,70,105,79,59]: n in [0..9] ]; // Vincenzo Librandi, Jun 28 2015

t={44}; While[n=t[[-1]]; s=If[EvenQ[n], 3*n/2, Round[3*n/4]]; Length[t]<100&&!MemberQ[t, s], AppendTo[t, s]]; t (* Vincenzo Librandi, Jun 28 2015 *)
PadRight[{},120,{44,66,99,74,111,83,62,93,70,105,79,59}] (* Harvey P. Dale, Jul 30 2025 *)

Vec(x*(44 + 66*x + 99*x^2 + 74*x^3 + 111*x^4 + 83*x^5 + 62*x^6 + 93*x^7 + 70*x^8 + 105*x^9 + 79*x^10 + 59*x^11) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)) + O(x^40)) \\ Colin Barker, Aug 16 2019

a(n+1) = A006368(a(n)).
From Colin Barker, Aug 16 2019: (Start)
G.f.: x*(44 + 66*x + 99*x^2 + 74*x^3 + 111*x^4 + 83*x^5 + 62*x^6 + 93*x^7 + 70*x^8 + 105*x^9 + 79*x^10 + 59*x^11) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)).
a(n) = a(n-12) for n>12.
(End)

A223088 Trajectory of 82 under the map n-> A006368(n).

Original entry on oeis.org

82, 123, 92, 138, 207, 155, 116, 174, 261, 196, 294, 441, 331, 248, 372, 558, 837, 628, 942, 1413, 1060, 1590, 2385, 1789, 1342, 2013, 1510, 2265, 1699, 1274, 1911, 1433, 1075, 806, 1209, 907, 680, 1020, 1530, 2295, 1721, 1291, 968, 1452, 2178, 3267, 2450, 3675

Offset: 1

N. J. A. Sloane, Mar 22 2013

nonn

It is conjectured that this trajectory does not close on itself.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.

Cf. A006369, A006368, A182205.

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

f:=n-> if n mod 2 = 0 then 3*n/2 elif n mod 4 = 1 then (3*n+1)/4 else (3*n-1)/4; fi;
t1:=[82];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {82}; While[n = t[[-1]]; s = If[EvenQ[n], 3*n/2, Round[3*n/4]]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> If[EvenQ[n], 3n/2, Round[3n/4]]}, {82}, 100] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

A349376 Dirichlet convolution of A006368 with the Dirichlet inverse of A006369, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

Original entry on oeis.org

1, 0, 0, 1, -3, 5, -4, -2, 1, 11, -7, -7, -7, 14, 7, 4, -10, 2, -11, -22, 10, 25, -14, 16, 7, 25, 0, -26, -17, -41, -18, -8, 17, 36, 34, 7, -21, 39, 17, 52, -24, -52, -25, -48, 1, 50, -28, -36, 8, -51, 24, -48, -31, 7, 62, 60, 27, 61, -35, 136, -35, 64, 0, 16, 62, -93, -39, -70, 34, -178, -42, -26, -42, 75, -27, -74

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Obviously, convolving this sequence with A006369 gives its inverse A006368 from n >= 1 onward.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A006369, A349368, A349377 (Dirichlet inverse), A349378 (sum with it).

Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349368 = Map();
A349368(n) = if(1==n,1,my(v); if(mapisdefined(memoA349368,n,&v), v, v = -sumdiv(n,d,if(dA006369(n/d)*A349368(d),0)); mapput(memoA349368,n,v); (v)));
A349376(n) = sumdiv(n,d,A006368(d)*A349368(n/d));

a(n) = Sum_{d|n} A006368(d) * A349368(n/d).

A349377 Dirichlet convolution of A006369 with the Dirichlet inverse of A006368, where A006368 is the "amusical permutation", and A006369 is its inverse permutation.

Original entry on oeis.org

1, 0, 0, -1, 3, -5, 4, 2, -1, -11, 7, 7, 7, -14, -7, -3, 10, -2, 11, 16, -10, -25, 14, -6, 2, -25, 0, 18, 17, 11, 18, 4, -17, -36, -10, 20, 21, -39, -17, -18, 24, 12, 25, 34, -7, -50, 28, 2, 8, -15, -24, 34, 31, 3, -20, -16, -27, -61, 35, 30, 35, -64, -8, -5, -20, 23, 39, 50, -34, 6, 42, -44, 42, -75, -15, 52, -22, 23

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Obviously, convolving this sequence with A006368 gives its inverse A006369 from n >= 1 onward.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A006369, A349351, A349376 (Dirichlet inverse), A349378 (sum with it).

Cf. also pairs A349613, A349614 and A349397, A349398 for similar constructions.

A006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
A006369(n) = if(!(n%3),(2/3)*n,(1/3)*if(1==(n%3),((4*n)-1),((4*n)+1)));
memoA349351 = Map();
A349351(n) = if(1==n,1,my(v); if(mapisdefined(memoA349351,n,&v), v, v = -sumdiv(n,d,if(dA006368(n/d)*A349351(d),0)); mapput(memoA349351,n,v); (v)));
A349377(n) = sumdiv(n,d,A006369(d)*A349351(n/d));

a(n) = Sum_{d|n} A006369(d) * A349351(n/d).

a(n) = A349378(n) - A349376(n).

A028397 Start at n and iterate the map in A006368; a(n) is the smallest number in the trajectory.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 4, 8, 8, 12, 8, 14, 8, 16, 8, 18, 14, 20, 16, 14, 8, 24, 14, 14, 20, 14, 14, 30, 8, 32, 14, 32, 14, 36, 14, 32, 14, 40, 8, 14, 32, 44, 32, 46, 14, 48, 14, 50, 32, 50, 40, 46, 8, 56, 32, 14, 44, 60, 46, 44, 14, 64, 14, 44, 50, 8, 50, 44, 40, 72, 8, 44, 56

Offset: 0

N. J. A. Sloane and J. H. Conway

			Sample iteration: 7->5->4->6->9->7 so a(7)=4.
Sample iteration: 12->18->27->20->30->45->34->51->... so a(12)=12.

Cf. A006368, A180853, A028393, A028395.

Table[Min[NestList[If[EvenQ[#],(3#)/2,Floor[(3#+2)/4]]&,n,100]],{n,0,80}] (* Harvey P. Dale, May 02 2012 *)

a(n)=local(m); if(n<=0,0,m=n; while((m!=n=(3*n+n%2)\(2+n%2*2))&n<10^99,m=min(m,n)); m)

$|=1; for($n=1;; ++$n){ $m=$n; $d{$m}=$n, $m=f($m) while !$d{$m};

if ($m<$n){ ($c,$m)=($d{$m},$n); $d{$m}=$c, $m=f($m) while $m >= $n }

print"$d{$n}," } sub f { $[0]%2 ? int((3*$[0]+1)/4) : 3*$_[0]/2 }

More terms from Hugo van der Sanden

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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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A180864 Trajectory of 13 under map n->A006368(n).

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A217218 Trajectory of 44 under the map k -> A006368(k).

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A223088 Trajectory of 82 under the map n-> A006368(n).

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