A354236 A(n,k) is the n-th number m such that the Collatz (or 3x+1) trajectory starting at m contains exactly k odd integers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..150, flattened

Wikipedia, Collatz Conjecture

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

Index entries for sequences that are permutations of the positive integers

Formula

A078719(A(n,k)) = k.

Example

Square array A(n,k) begins:
    1,   5,  3,  17, 11,  7,  9,  25,  33,  43, ...
    2,  10,  6,  34, 22, 14, 18,  49,  65,  86, ...
    4,  20, 12,  35, 23, 15, 19,  50,  66,  87, ...
    8,  21, 13,  68, 44, 28, 36,  51,  67,  89, ...
   16,  40, 24,  69, 45, 29, 37,  98, 130, 172, ...
   32,  42, 26,  70, 46, 30, 38,  99, 131, 173, ...
   64,  80, 48,  75, 88, 56, 72, 100, 132, 174, ...
  128,  84, 52, 136, 90, 58, 74, 101, 133, 177, ...
  256,  85, 53, 138, 92, 60, 76, 102, 134, 178, ...
  512, 160, 96, 140, 93, 61, 77, 196, 260, 179, ...

Maple

b:= proc(n) option remember; irem(n, 2, 'r')+
      `if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
    end:
A:= proc() local h, p, q; p, q:= proc() [] end, 0;
      proc(n, k)
        if k=1 then return 2^(n-1) fi;
        while nops(p(k))<n do q:= q+1;
          h:= b(q);
          p(h):= [p(h)[], q]
        od; p(k)[n]
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

b[n_] := b[n] = Module[{q, r}, {q, r} = QuotientRemainder[n, 2]; r +
     If[n == 1, 0, b[If[OddQ[n], 3*n + 1, q]]]];
A = Module[{h, p, q}, p[_] = {}; q = 0;
     Function[{n, k}, If[k == 1, 2^(n - 1)];
     While[Length[p[k]] < n, q = q + 1;
        h = b[q];
        p[h] = Append[p[h], q]];
     p[k][[n]]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jun 02 2022, after Alois P. Heinz *)

Crossrefs

Columns k=1-12 give: A011782, A062052, A062053, A062054, A062055, A062056, A062057, A062058, A062059, A062060, A072466, A072122.

Row n=1 gives A092893(k-1).

Cf. A006577, A006667, A078719.

Sequence in context: A261026 A160080 A026247 * A348779 A226693 A306300

Adjacent sequences: A354233 A354234 A354235 * A354237 A354238 A354239

Keyword

nonn,tabl

Author

Alois P. Heinz, May 20 2022

Status

approved