A380244 -id:A380244 - cp's OEIS frontend

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

Alois P. Heinz, May 20 2022

			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, ...

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

Columns k=1-12 give: A011782, A062052, A062053, A062054, A062055, A062056, A062057, A062058, A062059, A062060, A072466, A072122.
Row n=1 gives A092893(k-1).
Main diagonal gives A380244.
Cf. A006577, A006667, A078719.

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))

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 *)

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

A337144 n is the a(n)-th positive integer which takes its number of steps to reach 1 in the Collatz (or 3x+1) problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 1, 3, 1, 2, 3, 4, 1, 1, 1, 2, 3, 2, 3, 1, 2, 2, 1, 3, 2, 4, 5, 1, 2, 2, 1, 2, 3, 2, 3, 4, 1, 1, 1, 2, 5, 3, 4, 1, 2, 3, 5, 1, 2, 4, 5, 1, 2

Offset: 1

Alois P. Heinz, Jan 27 2021

			a(13) = 2 because A006577(13) = A006577(12) = 9 != A006577(j) for j < 12.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
Wikipedia, Collatz Conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A005186, A006577, A380244.

collatz:= proc(n) option remember; `if`(n=1, 0,
   1 + collatz(`if`(n::even, n/2, 3*n+1)))
end:
b:= proc() 0 end:
a:= proc(n) option remember; local t;
     `if`(n=1, 0, a(n-1));
      t:= collatz(n); b(t):= b(t)+1
    end:
seq(a(n), n=1..120);

collatz[n_] := collatz[n] = If[n == 1, 0,
   1 + collatz[If[EvenQ[n], n/2, 3n+1]]];
b[_] = 0;
a[n_] := a[n] = Module[{t},
   If[n == 1, 0, a[n-1]];
   t = collatz[n]; b[t] = b[t]+1];
Array[a, 120] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)

Ordinal transform of A006577.

a(n) = |{ j in {1..n} : A006577(j) = A006577(n) }|.

cp's OEIS Frontend

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.

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