A263716 Irregular triangle read by rows: numbers in the Collatz conjecture in the order of their first appearance.

1, 2, 3, 10, 5, 16, 8, 4, 6, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 9, 28, 14, 12, 15, 46, 23, 70, 35, 106, 53, 160, 80, 18, 19, 58, 29, 88, 44, 21, 64, 32, 24, 25, 76, 38, 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182

Offset

0,2

Comments

This is the irregular triangle read by rows giving trajectory of n in the Collatz problem, flattened and with all the repeated terms deleted.

This sequence goes to infinity as n gets larger. On the Collatz conjecture this sequence is a permutation of the positive integers. [Corrected by Charles R Greathouse IV, Jul 29 2016]

Links

Daniel Suteu, Table of n, a(n) for n = 0..19999

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

Formula

row(n) = {

if seen[n]: stop

else: write(n) and do:

| n is one: stop

| n is odd: n <- 3*n+1

| n is even: n <- n/2

}

Example

Triangle begins:
1;
2;
3, 10, 5, 16, 8, 4;
...
The Collatz trajectories for the first five positive integers are {1}, {2, 1}, {3, 10, 5, 16, 8, 4, 2, 1}, {4, 2, 1}, {5, 16, 8, 4, 2, 1}.
From {2, 1} we delete 1 because it has already occurred. From {3, 10, 5, ..., 4, 2, 1} we delete {2, 1} because both numbers have already occurred. We completely get rid of {4, 2, 1} because it has already occurred as the tail end of {3, 10, 5, ...}, and we also completely get rid of {5, 16, 8, ...} for the same reason.
This leaves us with {1}, {2}, {3, 10, 5, 16, 8, 4}, thus accounting for the first eight terms of this sequence.

Mathematica

collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; DeleteDuplicates[Flatten[Table[collatz[n], {n, 20}]]] (* Alonso del Arte, Oct 24 2015 *)

Prog

(Sidef)
func collatz(n) is cached {  # automatically memoized function
    say n;                   # prints the first unseen numbers
    n.is_one ? 0
             : (n.is_even ? collatz(n/2)
                          : collatz(3*n + 1));
}
range(1, Math.inf).each { |i| collatz(i) }

Crossrefs

Cf. A006577, A070165, A222118 (row lengths).

Cf. A347265 (essentially the same).

Sequence in context: A182076 A347271 A266552 * A344457 A175899 A328613

Adjacent sequences: A263713 A263714 A263715 * A263717 A263718 A263719

Keyword

nonn,tabf,easy,nice

Author

Daniel Suteu, Oct 24 2015

Status

approved