A006369 -id:A006369 - cp's OEIS frontend

A094328 Iterate the map in A006369 starting at 4.

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

Offset: 1

N. J. A. Sloane, Jun 04 2004

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 270.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
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. A006368, A028394-A028397, A094329, A185589, A185590.

a094328 n = a094328_list !! (n-1)
a094328_list = iterate a006369 4  -- Reinhard Zumkeller, Dec 31 2011

Table[{4, 5, 7, 9, 6}, {21}] // Flatten  (* Jean-François Alcover, Jun 10 2013 *)
LinearRecurrence[{0, 0, 0, 0, 1},{4, 5, 7, 9, 6},105] (* Ray Chandler, Sep 03 2015 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,0,0,0]^(n-1)*[4;5;7;9;6])[1,1] \\ Charles R Greathouse IV, Oct 18 2022

The map is: n -> if n mod 3 = 0 then 2*n/3 elif n mod 3 = 1 then (4*n-1)/3 else (4*n+1)/3.

Periodic with period length 5.

A028394 Iterate the map in A006369 starting at 8.

Original entry on oeis.org

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

Offset: 0

J. H. Conway

nonn

It is an unsolved problem to determine if this sequence is bounded or unbounded.

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 270.

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.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006369, A028396, A094328, A094329, A185589, A185590.

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

a028394 n = a028394_list !! n
a028394_list = iterate a006369 8  -- Reinhard Zumkeller, Dec 31 2011

G := proc(n) option remember; if n = 0 then 8 elif 4*G(n-1) mod 3 = 0 then 2*G(n-1)/3 else round(4*G(n-1)/3); fi; end; [ seq(G(i),i=0..80) ];
f:=proc(N) local 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; # N. J. A. Sloane, Feb 04 2011

nxt[n_]:=Module[{m=Mod[n,3]},Which[m==0,(2n)/3,m==1,(4n-1)/3,True,(4n+1)/3]]; NestList[nxt,8,60] (* Harvey P. Dale, Dec 13 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n-1)/3, , (4n+1)/3 ] }, {8}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

The map is: n -> if n mod 3 = 0 then 2*n/3 elif n mod 3 = 1 then (4*n-1)/3 else (4*n+1)/3.

A094329 Iterate the map in A006369 starting at 16.

Original entry on oeis.org

16, 21, 14, 19, 25, 33, 22, 29, 39, 26, 35, 47, 63, 42, 28, 37, 49, 65, 87, 58, 77, 103, 137, 183, 122, 163, 217, 289, 385, 513, 342, 228, 152, 203, 271, 361, 481, 641, 855, 570, 380, 507, 338, 451, 601, 801, 534, 356, 475, 633, 422, 563, 751, 1001, 1335, 890, 1187, 1583

Offset: 1

N. J. A. Sloane, Jun 04 2004

nonn

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 270.

See A006368, A028394, A028396, A094328, A185589, A185590.

a094329 n = a094329_list !! (n-1)
a094329_list = iterate a006369 16  -- Reinhard Zumkeller, Dec 31 2011

SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {16}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

A185589 Iterate the map in A006369 starting at 144.

Original entry on oeis.org

144, 96, 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, 3870, 2580, 1720, 2293, 3057, 2038, 2717, 3623, 4831

Offset: 1

N. J. A. Sloane, Feb 04 2011

nonn

Lagarias, page 270, appears to imply that this trajectory has period 12 and smallest element 144, which is not true.

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 270.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A028394, A028396, A094328, A094329, A185590.

a185589 n = a185589_list !! (n-1)
a185589_list = iterate a006369 144  -- Reinhard Zumkeller, Dec 31 2011

f[n_] := If[Mod[n, 3] == 0, 2*n/3, Round[4*n/3]]; a[1] = 144; a[n_] := a[n] = f[a[n - 1]]; Table[a[n], {n, 1, 59}]  (* Jean-François Alcover, Jun 10 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {144}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

A185590 Iterate the map in A006369 starting at 44.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 04 2011

Periodic with period length 12.

A. O. L. Atkin, Comment on Problem 63-13, SIAM Rev., 8 (1966), 234-236.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 270. (beware of typo: Lagarias says the orbit of 144 (not 44) has period 12.)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006369, A028394, A028396, A094328, A094329, A185589.

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

a185590 n = a185590_list !! (n-1)
a185590_list = iterate a006369 44  -- Reinhard Zumkeller, Dec 31 2011

f[n_] := If[Mod[n, 3] == 0, 2*n/3, Round[4*n/3]]; a[1] = 44; a[n_] := a[n] = f[a[n - 1]]; Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jun 10 2013 *)

A028396 Iterate the map in A006369 starting at 14.

Original entry on oeis.org

14, 19, 25, 33, 22, 29, 39, 26, 35, 47, 63, 42, 28, 37, 49, 65, 87, 58, 77, 103, 137, 183, 122, 163, 217, 289, 385, 513, 342, 228, 152, 203, 271, 361, 481, 641, 855, 570, 380, 507, 338, 451, 601, 801, 534, 356, 475, 633, 422, 563, 751, 1001, 1335, 890, 1187

Offset: 0

J. H. Conway

nonn

Cf. A028394, A028396, A094328, A094329, A185589, A185590.

a028396 n = a028396_list !! n
a028396_list = iterate a006369 14  -- Reinhard Zumkeller, Dec 31 2011

SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {14}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

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 *)

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).

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A094328 Iterate the map in A006369 starting at 4.

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A028394 Iterate the map in A006369 starting at 8.

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A094329 Iterate the map in A006369 starting at 16.

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A185589 Iterate the map in A006369 starting at 144.

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A185590 Iterate the map in A006369 starting at 44.

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A028396 Iterate the map in A006369 starting at 14.

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

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

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