A375888 Rectangular array: row n shows all k such that n is the number of rises in the trajectory of k in the Collatz problem.

1, 2, 5, 4, 10, 3, 8, 20, 6, 17, 16, 21, 12, 34, 11, 32, 40, 13, 35, 22, 7, 64, 42, 24, 68, 23, 14, 9, 128, 80, 26, 69, 44, 15, 18, 25, 256, 84, 48, 70, 45, 28, 19, 49, 33, 512, 85, 52, 75, 46, 29, 36, 50, 65, 43, 1024, 160, 53, 136, 88, 30, 37, 51, 66, 86, 57

Offset

0,2

Comments

Assuming that the Collatz conjecture (also known as the 3x+1 conjecture) is true, this is a permutation of the positive integers; viz., every positive integer occurs exactly once. Conjecture: every row contains a pair of consecutive integers.

Links

Table of n, a(n) for n=0..65.

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

Transpose of the array in A354236.

Example

Corner:
   1     2     4     8    16    32    64   128   256   512  1024
   5    10    20    21    40    42    80    84    85   160   168
   3     6    12    13    24    26    48    52    53    96   104
  17    34    35    68    69    70    75   136   138   140   141
  11    22    23    44    45    46    88    90    92    93   176
   7    14    15    28    29    30    56    58    60    61   112
   9    18    19    36    37    38    72    74    76    77    81
6 is in row 2 because the trajectory, (6, 3, 10, 5, 16, 4, 2, 1), has exactly 2 rises: 3 to 10, and 5 to 16.

Mathematica

t = Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]], _? Positive], {n, 2048}]; (* after Harvey P. Dale, A006667 *)
r[n_] := Flatten[Position[t, n - 1]];
Column[Table[r[n], {n, 1, 21}]] (* array *)
u = Table[r[k][[n + 1 - k]], {n, 1, 12}, {k, 1, n}]
Flatten[u] (* sequence *)

Crossrefs

Cf. A000027, A000079 (row 1), A092893 (column 1), A006667, A070265, A078719.

Cf. A354236.

Sequence in context: A324142 A339809 A360896 * A084432 A071297 A356875

Adjacent sequences: A375885 A375886 A375887 * A375889 A375890 A375891

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 11 2024

Status

approved