A347265 -id:A347265 - cp's OEIS frontend

A347270 Square array T(n,k) in which row n lists the 3x+1 sequence starting at n, read by antidiagonals upwards, with n >= 1 and k >= 0.

1, 2, 4, 3, 1, 2, 4, 10, 4, 1, 5, 2, 5, 2, 4, 6, 16, 1, 16, 1, 2, 7, 3, 8, 4, 8, 4, 1, 8, 22, 10, 4, 2, 4, 2, 4, 9, 4, 11, 5, 2, 1, 2, 1, 2, 10, 28, 2, 34, 16, 1, 4, 1, 4, 1, 11, 5, 14, 1, 17, 8, 4, 2, 4, 2, 4, 12, 34, 16, 7, 4, 52, 4, 2, 1, 2, 1, 2, 13, 6, 17, 8, 22

Offset: 1

Omar E. Pol, Aug 25 2021

This array gives all 3x+1 sequences.
The 3x+1 or Collatz problem is described in A006370.
Column k gives the image of n at the k-th step.
This infinite square array contains the irregular triangles A070165, A235795 and A347271.
For a piping diagram of the 3x+1 problem see A235800.

			The corner of the square array begins:
   1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
   2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
   3,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
   4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...
   5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...
   6, 3,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, ...
   7,22,11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, 1, ...
   8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
   9,28,14, 7,22,11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, ...
  10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
  11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, ...
  12, 6, 3,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
  13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, ...
  14, 7,22,11,34,17,52,26,13,40,20,10, 5,16, 8, 4, 2, 1, 4, 2, ...
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
J. C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.
Index entries for sequences related to 3x+1 (or Collatz) problem

Main diagonal gives A347272.
Parity of this sequence is A347283.
Largest value in row n gives A056959.
Number of nonpowers of 2 in row n gives A208981.
Some rows n are: A153727 (n=1), A033478 (n=3), A033479 (n=9), A033480 (n=15), A033481 (n=21), A008884 (n=27), A008880 (n=33), A008878 (n=39), A008883 (n=51), A008877 (n=57), A008874 (n=63), A258056 (n=75), A258098 (n=79), A008876 (n=81), A008879 (n=87), A008875 (n=95), A008873 (n=97), A008882 (n=99), A245671 (n=1729).
First four columns k are: A000027 (k=0), A006370 (k=1), A075884 (k=2), A076536 (k=3).
Cf. A006877, A014682, A057716, A070165, A078719, A135282, A235795, A235800, A235801, A347265, A347267, A347268, A347269, A347271, A347519.

T:= proc(n, k) option remember; `if`(k=0, n, (j->
      `if`(j::even, j/2, 3*j+1))(T(n, k-1)))
    end:
seq(seq(T(d-k, k), k=0..d-1), d=1..20);  # Alois P. Heinz, Aug 25 2021

T[n_, k_] := T[n, k] = If[k == 0, n, Function[j,
     If[EvenQ[j], j/2, 3*j + 1]][T[n, k - 1]]];
Table[Table[T[d - k, k], {k, 0, d - 1}], {d, 1, 20}] // Flatten (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

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

Original entry on oeis.org

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

Daniel Suteu, Oct 24 2015

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]

			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.

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

Cf. A347265 (essentially the same).

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

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) }

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
}

cp's OEIS Frontend

A347270 Square array T(n,k) in which row n lists the 3x+1 sequence starting at n, read by antidiagonals upwards, with n >= 1 and k >= 0.

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