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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.

%I A375888 #13 Sep 12 2024 20:00:58
%S A375888 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,
%T A375888 23,14,9,128,80,26,69,44,15,18,25,256,84,48,70,45,28,19,49,33,512,85,
%U A375888 52,75,46,29,36,50,65,43,1024,160,53,136,88,30,37,51,66,86,57
%N 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.
%C A375888 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.
%H A375888 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A375888 Transpose of the array in A354236.
%e A375888 Corner:
%e A375888    1     2     4     8    16    32    64   128   256   512  1024
%e A375888    5    10    20    21    40    42    80    84    85   160   168
%e A375888    3     6    12    13    24    26    48    52    53    96   104
%e A375888   17    34    35    68    69    70    75   136   138   140   141
%e A375888   11    22    23    44    45    46    88    90    92    93   176
%e A375888    7    14    15    28    29    30    56    58    60    61   112
%e A375888    9    18    19    36    37    38    72    74    76    77    81
%e A375888 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.
%t A375888 t = Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]], _? Positive], {n, 2048}]; (* after _Harvey P. Dale_, A006667 *)
%t A375888 r[n_] := Flatten[Position[t, n - 1]];
%t A375888 Column[Table[r[n], {n, 1, 21}]] (* array *)
%t A375888 u = Table[r[k][[n + 1 - k]], {n, 1, 12}, {k, 1, n}]
%t A375888 Flatten[u] (* sequence *)
%Y A375888 Cf. A000027, A000079 (row 1), A092893 (column 1), A006667, A070265, A078719.
%Y A375888 Cf. A354236.
%K A375888 nonn,tabl
%O A375888 0,2
%A A375888 _Clark Kimberling_, Sep 11 2024