A006368 -id:A006368 - cp's OEIS frontend

A168221 a(n) = A006368(A006368(n)).

0, 1, 2, 3, 9, 6, 7, 4, 18, 5, 11, 12, 27, 15, 16, 8, 36, 10, 20, 21, 45, 24, 25, 13, 54, 14, 29, 30, 63, 33, 34, 17, 72, 19, 38, 39, 81, 42, 43, 22, 90, 23, 47, 48, 99, 51, 52, 26, 108, 28, 56, 57, 117, 60, 61, 31, 126, 32, 65, 66, 135, 69, 70, 35, 144, 37, 74, 75, 153, 78, 79, 40

Offset: 0

Reinhard Zumkeller, Nov 20 2009

Inverse integer permutation to A168222;

a(A006369(n)) = A006368(n).

A.H.M. Smeets, Table of n, a(n) for n = 0..20000
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1).

Cf. A006368, A006369, A168222.

Table[Nest[If[OddQ[#],Floor[(3#+2)/4],3#/2]&,n,2],{n,0,100}] (* Paolo Xausa, Dec 15 2023 *)
LinearRecurrence[{0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1},{0,1,2,3,9,6,7,4,18,5,11,12,27,15,16,8,36,10,20,21,45,24,25,13},80] (* Harvey P. Dale, Feb 07 2024 *)

def A006368(n):
    if n%2 == 0:
        return 3*(n//2)
    elif n%4 == 1:
        return 3*(n//4)+1
    else:
        return 3*(n+1)//4-1
n = 0
while n < 30:
    print(n,A006368(A006368(n)))
    n = n+1 # A.H.M. Smeets, Aug 14 2019

From Luce ETIENNE, Aug 14 2019: (Start)
a(n) = 2*a(n-16) - a(n-32).
a(n) = (-18*(40*m^7 - 973*m^6 + 9352*m^5 - 45115*m^4 + 114520*m^3 - 145432*m^2 + 75168*m - 10080)*floor(n/8) - m*(332*m^6 - 7973*m^5 + 75236*m^4 - 352835*m^3 + 855008*m^2 - 999992*m + 422664) + m*(4*m^6 - 105*m^5 + 1120*m^4 - 6195*m^3 + 18676*m^2 - 28980*m + 18000)*(-1)^(n/8))/10080 where m = n mod 8.
(End)
From A.H.M. Smeets, Aug 14 2019: (Start)
a(4*n) = 9*n.
a(8*n+1) = a(8*n-1)+1, n > 0.
a(8*n+3) = a(8*n+2)+1.
a(8*n+5) = a(8*n+3)+3 = a(8*n+2)+4.
a(8*n+6) = a(8*n+5)+1 = a(8*n+3)+4 = a(8*n+2)+5.
a(16*n+1) = 9*n+1.
a(16*n+2) = 18*n+2.
a(16*n+3) = a(16*n+2)+1 = 18*n+3.
a(16*n+5) = a(16*n+3)+3 = 18*n+6.
a(16*n+6) = a(16*n+5)+1 = 18*n+7.
a(16*n+7) = (a(16*n+6)+1)/2 = 9*n+4.
a(16*n+9) = 9*n+5.
a(16*n+10) = 2*a(16*n+9)+1 = 18*n+11.
a(16*n+11) = a(16*n+10)+1 = 18*n+12.
a(16*n+13) = a(16*n+11)+3 = 18*n+15.
a(16*n+14) = a(16*n+13) = 18*n+16.
a(16*n+15) = a(16*n+14)/2 = 9*n+8.
From this, (9*n-7)/16 <= a(n) <= 9*n/4.
(End)
From Colin Barker, Aug 23 2019: (Start)
G.f.: x*(1 - x + x^2)*(1 + 3*x + 5*x^2 + 11*x^3 + 12*x^4 + 8*x^5 + 10*x^7 + 14*x^8 + 13*x^9 + 8*x^10 + 13*x^11 + 14*x^12 + 10*x^13 + 8*x^15 + 12*x^16 + 11*x^17 + 5*x^18 + 3*x^19 + x^20) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2*(1 + x^4)^2*(1 + x^8)).
a(n) = a(n-8) + a(n-16) - a(n-24) for n>23.
(End)

A168223 a(n) = A006369(n) - A006368(n).

Original entry on oeis.org

0, 0, 0, 0, -1, 3, -5, 4, -1, -1, -2, 7, -10, 7, -2, -1, -3, 10, -15, 11, -3, -2, -4, 14, -20, 14, -4, -2, -5, 17, -25, 18, -5, -3, -6, 21, -30, 21, -6, -3, -7, 24, -35, 25, -7, -4, -8, 28, -40, 28, -8, -4, -9, 31, -45, 32, -9, -5, -10, 35, -50, 35, -10, -5, -11, 38, -55, 39

Offset: 0

Reinhard Zumkeller, Nov 20 2009

A047342 and A168223 give record values and where they occur: a(A168224(n))=A047342(n) and a(m) < A047342(n) for m < A168224(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (-2,-2,0,3,4,3,0,-2,-2,-1).

a168223 n = a006369 n - a006368 n  -- Reinhard Zumkeller, Mar 15 2014

LinearRecurrence[{-2,-2,0,3,4,3,0,-2,-2,-1},{0, 0, 0, 0, -1, 3, -5, 4, -1, -1},50] (* G. C. Greubel, Jul 16 2016 *)

a(12*n) = -10*n,          a(12*n+1) = 7*n.
a(12*n+2) = -2*n,         a(12*n+3) = -n.
a(12*n+4) = -2*n - 1,     a(12*n+5) = 7*n + 3.
a(12*n+6) = -10*n - 5,    a(12*n+7) = 7*n + 4.
a(12*n+8) = -2*n -1,      a(12*n+9) = -n - 1.
a(12*n+10) = -2*n - 2,    a(12*n+11) = 7*n + 7.
G.f.: -x^4*(x^2-x+1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^2+x+1)^2). - Colin Barker, Apr 04 2013

A028398 When map in A006368 is iterated, all numbers fall into cycles; order cycles by smallest entry; a(n) is smallest entry in n-th cycle (some cycles are infinite).

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 40, 44, 64, 80, 82, 104, 136, 172, 184, 188, 242, 256, 274, 280, 296, 352, 368, 382, 386, 424, 472, 496, 526, 530, 608, 622, 638, 640, 652, 670, 688, 692, 712, 716, 752, 760, 782, 784, 800, 814, 824, 832, 860, 878, 904, 910, 932, 964, 980, 1022

Offset: 0

J. H. Conway and Wouter Meeussen

nonn

Iterations of A006368 starting with a(3)=4, a(4)=8, a(5)=14 and a(6)=40 give trajectories A180853, A028393, A028395, A182205 respectively. [Reinhard Zumkeller, Apr 18 2012]

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

A349351 Dirichlet inverse of A006368, "the amusical permutation", a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

Original entry on oeis.org

1, -3, -2, 3, -4, 3, -5, -3, -3, 9, -8, 6, -10, 9, 5, 3, -13, 15, -14, 0, 4, 15, -17, -15, -3, 21, 0, 9, -22, 9, -23, -3, 7, 27, 14, -24, -28, 27, 11, -9, -31, 27, -32, 12, 18, 33, -35, 24, -12, 15, 14, 6, -40, 9, 23, -27, 13, 45, -44, -63, -46, 45, 27, 3, 31, 39, -50, 9, 16, 9, -53, 6, -55, 57, 12, 18, 22, 33, -59

Offset: 1

Antti Karttunen, Nov 15 2021

sign

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

Cf. A006368, A349352, A349368, A349377.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
v349351 = DirInverseCorrect(vector(up_to,n,A006368(n)));
A349351(n) = v349351[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A006368(n/d) * a(d).

a(n) = A349352(n) - A006368(n).

A349352 Sum of A006368, "the amusical permutation", and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 9, 4, 24, 0, 24, 0, 30, 16, 27, 0, 42, 0, 30, 20, 48, 0, 21, 16, 60, 20, 51, 0, 54, 0, 45, 32, 78, 40, 30, 0, 84, 40, 51, 0, 90, 0, 78, 52, 102, 0, 96, 25, 90, 52, 84, 0, 90, 64, 57, 56, 132, 0, 27, 0, 138, 74, 99, 80, 138, 0, 111, 68, 114, 0, 114, 0, 168, 68, 132, 80, 150, 0, 138, 61, 186, 0

Offset: 1

Antti Karttunen, Nov 15 2021

sign

The first negative term is a(2520) = -918.

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

Cf. A006368, A349351, A349369, A349378.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA006368(n) = ((3*n)+(n%2))\(2+((n%2)*2));
v349351 = DirInverseCorrect(vector(up_to,n,A006368(n)));
A349351(n) = v349351[n];
A349352(n) = (A006368(n)+A349351(n));

a(n) = A006368(n) + A349351(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A006368(d) * A349351(n/d).

A349369 Sum of A006369 and its Dirichlet inverse, where A006369 is the inverse of "amusical permutation", A006368.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 3, 4, 42, 0, -10, 0, 54, 28, 37, 0, 16, 0, -41, 36, 90, 0, 60, 49, 102, 16, -39, 0, -84, 0, 11, 60, 138, 126, 30, 0, 150, 68, 221, 0, -92, 0, -81, 40, 186, 0, -72, 81, -61, 92, -79, 0, 36, 210, 259, 100, 234, 0, 394, 0, 246, 56, 149, 238, -172, 0, -121, 124, -352, 0, 8, 0, 294, -22, -119, 270

Offset: 1

Antti Karttunen, Nov 17 2021

sign

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

Cf. A006369, A349368.

Cf. also A349352, A349378.

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)));
A349369(n) = (A006369(n)+A349368(n));

a(n) = A006369(n) + A349368(n).

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A006369(d) * A349368(n/d).

A368179 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006368 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 6, 3, 3, 1, 0, 6, 4, 9, 2, 2, 1, 0, 7, 9, 6, 7, 3, 3, 1, 0, 8, 5, 7, 9, 5, 2, 2, 1, 0, 9, 12, 4, 5, 7, 4, 3, 3, 1, 0, 10, 7, 18, 6, 4, 5, 6, 2, 2, 1, 0, 11, 15, 5, 27, 9, 6, 4, 9, 3, 3, 1, 0, 12, 8, 11, 4, 20, 7, 9, 6, 7, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 15 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,  6,  9,  7,  5,  4,  6,  9,  7,  5,  4, ... = A180853
  [ 5]   5,  4,  6,  9,  7,  5,  4,  6,  9,  7,  5, ... = A180853 (shifted)
  [ 6]   6,  9,  7,  5,  4,  6,  9,  7,  5,  4,  6, ... = A180853 (shifted)
  [ 7]   7,  5,  4,  6,  9,  7,  5,  4,  6,  9,  7, ... = A180853 (shifted)
  [ 8]   8, 12, 18, 27, 20, 30, 45, 34, 51, 38, 57, ... = A028393
  [ 9]   9,  7,  5,  4,  6,  9,  7,  5,  4,  6,  9, ... = A180853 (shifted)
  [10]  10, 15, 11,  8, 12, 18, 27, 20, 30, 45, 34, ... = A180864 (shifted)
  ...    |   |   |
      A001477|A168221
             |
          A006368

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,  A028393, A028395, A094332, A176059, A180853, A180864, A182205, A217218, A223086, A223087, A223088.
Columns: A001477, A006368, A168221.
Main diagonal: A368180.

A006368[n_]:=If[OddQ[n],Floor[(3n+2)/4],3n/2];
A368179list[dmax_]:=With[{a=Reverse[Table[NestList[A006368,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368179list[15] (* Generates 15 antidiagonals *)

A368180 Main diagonal of A368179: the n-th term in the trajectory of n under the A006368 map.

Original entry on oeis.org

0, 1, 2, 2, 5, 5, 9, 4, 51, 6, 34, 57, 48, 38, 132, 48, 167, 32, 205, 167, 130, 106, 167, 243, 50, 159, 125, 369, 297, 119, 702, 130, 844, 84, 500, 50, 1215, 119, 1424, 142, 840, 312, 126, 3041, 70, 7209, 22143, 540, 1623, 160, 15165, 1443, 8867, 3406, 10509, 1899, 12146, 578, 911, 79

Offset: 0

Paolo Xausa, Dec 15 2023

Cf. A006368, A368179.

A368180[n_]:=Nest[If[OddQ[#],Floor[(3#+2)/4],3#/2]&,n,n];
Array[A368180,100,0]

a(n) = A368179(n,n).

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.

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 *)

cp's OEIS Frontend

A168221 a(n) = A006368(A006368(n)).

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A168223 a(n) = A006369(n) - A006368(n).

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A028398 When map in A006368 is iterated, all numbers fall into cycles; order cycles by smallest entry; a(n) is smallest entry in n-th cycle (some cycles are infinite).

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A349351 Dirichlet inverse of A006368, "the amusical permutation", a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

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A349352 Sum of A006368, "the amusical permutation", and its Dirichlet inverse.

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A349369 Sum of A006369 and its Dirichlet inverse, where A006369 is the inverse of "amusical permutation", A006368.

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A368179 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006368 map.

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A368180 Main diagonal of A368179: the n-th term in the trajectory of n under the A006368 map.

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A094332 Iterate the map in A006368 starting at 12.

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