A094328 -id:A094328 - cp's OEIS frontend

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

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)

A217729 Trajectory of 40 under the map n-> A006369(n).

Original entry on oeis.org

40, 53, 71, 95, 127, 169, 225, 150, 100, 133, 177, 118, 157, 209, 279, 186, 124, 165, 110, 147, 98, 131, 175, 233, 311, 415, 553, 737, 983, 1311, 874, 1165, 1553, 2071, 2761, 3681, 2454, 1636, 2181, 1454, 1939, 2585, 3447, 2298, 1532, 2043, 1362, 908, 1211, 1615

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:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[40];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {40}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {40}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

A223083 Trajectory of 64 under the map n-> A006369(n).

Original entry on oeis.org

64, 85, 113, 151, 201, 134, 179, 239, 319, 425, 567, 378, 252, 168, 112, 149, 199, 265, 353, 471, 314, 419, 559, 745, 993, 662, 883, 1177, 1569, 1046, 1395, 930, 620, 827, 1103, 1471, 1961, 2615, 3487, 4649, 6199, 8265, 5510, 7347, 4898, 6531, 4354, 5805

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:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[64];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {64}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {64}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

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 *)

A094332 Iterate the map in A006368 starting at 12.

Original entry on oeis.org

12, 8, 11, 15, 10, 13, 17, 23, 31, 41, 55, 73, 97, 129, 86, 115, 153, 102, 68, 91, 121, 161, 215, 287, 383, 511, 681, 454, 605, 807, 538, 717, 478, 637, 849, 566, 755, 1007, 1343, 1791, 1194, 796, 1061, 1415, 1887, 1258, 1677, 1118, 1491, 994, 1325, 1767, 1178, 1571

Offset: 1

N. J. A. Sloane, Jun 04 2004

nonn

See A028384, A006368, A094328, etc. for more information.

A223084 Trajectory of 80 under the map n-> A006369(n).

Original entry on oeis.org

80, 107, 143, 191, 255, 170, 227, 303, 202, 269, 359, 479, 639, 426, 284, 379, 505, 673, 897, 598, 797, 1063, 1417, 1889, 2519, 3359, 4479, 2986, 3981, 2654, 3539, 4719, 3146, 4195, 5593, 7457, 9943, 13257, 8838, 5892, 3928, 5237, 6983, 9311, 12415, 16553, 22071

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:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[80];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {80}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {80}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

A223085 Trajectory of 82 under the map n-> A006369(n).

Original entry on oeis.org

82, 109, 145, 193, 257, 343, 457, 609, 406, 541, 721, 961, 1281, 854, 1139, 1519, 2025, 1350, 900, 600, 400, 533, 711, 474, 316, 421, 561, 374, 499, 665, 887, 1183, 1577, 2103, 1402, 1869, 1246, 1661, 2215, 2953, 3937, 5249, 6999, 4666, 6221, 8295, 5530, 7373

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:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
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 = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {82}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)
NestList[If[Divisible[#,3],(2#)/3,Floor[(4#)/3+1/2]]&,82,50] (* Harvey P. Dale, Sep 22 2019 *)

A223086 Trajectory of 64 under the map n-> A006368(n).

Original entry on oeis.org

64, 96, 144, 216, 324, 486, 729, 547, 410, 615, 461, 346, 519, 389, 292, 438, 657, 493, 370, 555, 416, 624, 936, 1404, 2106, 3159, 2369, 1777, 1333, 1000, 1500, 2250, 3375, 2531, 1898, 2847, 2135, 1601, 1201, 901, 676, 1014, 1521, 1141, 856, 1284, 1926, 2889

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:=[64];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {64}; 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]]}, {64}, 100] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

A223087 Trajectory of 80 under the map n-> A006368(n).

Original entry on oeis.org

80, 120, 180, 270, 405, 304, 456, 684, 1026, 1539, 1154, 1731, 1298, 1947, 1460, 2190, 3285, 2464, 3696, 5544, 8316, 12474, 18711, 14033, 10525, 7894, 11841, 8881, 6661, 4996, 7494, 11241, 8431, 6323, 4742, 7113, 5335, 4001, 3001, 2251, 1688, 2532, 3798, 5697

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:=[80];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {80}; 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]]}, {80}, 100] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

A368227 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006369 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 5, 3, 3, 1, 0, 6, 7, 7, 2, 2, 1, 0, 7, 4, 9, 9, 3, 3, 1, 0, 8, 9, 5, 6, 6, 2, 2, 1, 0, 9, 11, 6, 7, 4, 4, 3, 3, 1, 0, 10, 6, 15, 4, 9, 5, 5, 2, 2, 1, 0, 11, 13, 4, 10, 5, 6, 7, 7, 3, 3, 1, 0, 12, 15, 17, 5, 13, 7, 4, 9, 9, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 18 2023

			Array begins:
  [ 0]   0,  0,  0,  0,  0,  0,  0,  0,  0,   0,  0, ... = A000004
  [ 1]   1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1, ... = A000012
  [ 2]   2,  3,  2,  3,  2,  3,  2,  3,  2,   3,  2, ... = A010693
  [ 3]   3,  2,  3,  2,  3,  2,  3,  2,  3,   2,  3, ... = A176059
  [ 4]   4,  5,  7,  9,  6,  4,  5,  7,  9,   6,  4, ... = A094328
  [ 5]   5,  7,  9,  6,  4,  5,  7,  9,  6,   4,  5, ... = A094328 (shifted)
  [ 6]   6,  4,  5,  7,  9,  6,  4,  5,  7,   9,  6, ... = A094328 (shifted)
  [ 7]   7,  9,  6,  4,  5,  7,  9,  6,  4,   5,  7, ... = A094328 (shifted)
  [ 8]   8, 11, 15, 10, 13, 17, 23, 31, 41,  55, 73, ... = A028394
  [ 9]   9,  6,  4,  5,  7,  9,  6,  4,  5,   7,  9, ... = A094328 (shifted)
  [10]  10, 13, 17, 23, 31, 41, 55, 73, 97, 129, 86, ... = A028394 (shifted)
  ...    |   |   |
      A001477|A168222
          A006369

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem

Rows: A000004, A000012, A010693, A028394, A028396, A094328, A094329, A176059, A185589, A185590, A217729, A223083, A223084, A223085.
Columns: A001477, A006369, A168222.
Main diagonal: A368228.
Cf. A368179.

A006369[n_]:=If[Divisible[n,3],2n/3,Round[4n/3]];
A368227list[dmax_]:=With[{a=Reverse[Table[NestList[A006369,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368227list[15] (* Generates 15 antidiagonals *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A217729 Trajectory of 40 under the map n-> A006369(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A223083 Trajectory of 64 under the map n-> A006369(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A094332 Iterate the map in A006368 starting at 12.

Views

Author

Keywords

Links

Crossrefs

A223084 Trajectory of 80 under the map n-> A006369(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A223085 Trajectory of 82 under the map n-> A006369(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A223086 Trajectory of 64 under the map n-> A006368(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple