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

Author

Daniel Suteu, Oct 24 2015

Keywords

• nonn
• tabf
• easy
• nice

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]

Examples

			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.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 0..19999
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

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

Cf. A347265 (essentially the same).

Programs

Mathematica

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

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) }

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

}