This site is supported by donations to The OEIS Foundation.

# 3x-1 problem

The
 3 x  −  1
problem
is a variation on the 3 x + 1 problem (or Collatz problem).
The Collatz problem, named after Lothar Collatz who first proposed it in 1937, asks whether the iterated Collatz function
 f  (n)
always reaches 1 when
 n > 0
, with the Collatz function defined as
${\displaystyle f(n)={\begin{cases}n/2&{\mbox{if }}n\equiv 0{\pmod {2}},\\3n+1&{\mbox{if }}n\equiv 1{\pmod {2}}.\end{cases}}\,}$
The
 3 x  −  1
problem instead uses the iterated function
${\displaystyle f(n)={\begin{cases}n/2&{\mbox{if }}n\equiv 0{\pmod {2}},\\3n-1&{\mbox{if }}n\equiv 1{\pmod {2}}.\end{cases}}\,}$

## The 3 x  −  1 problem trajectories

The
 3 x  −  1
problem trajectories might be of 3 kinds
• Trajectories that eventually become nontrivially (1, 2, 1, 2, 1, 2, ... excluded) cyclic;
• Trajectories that eventually hit a power of 2 (thus falling straight down to 1, like hailstones, and then going through the trivial cycle 1, 2, 1, 2, 1, 2, ...).

Do
 3 x  −  1
problem trajectories heading towards infinity exist or not?

### Eventually nontrivially cyclic trajectories

The
 3 x  −  1
problem does have eventually nontrivially cyclic trajectories!
A003124 One of the basic cycles (of length 18) in the
 x ↦ 3 x  −  1
(
 x
odd) or
 x / 2
(
 x
even) problem.
{17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34}
&
{17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34}
& ...

### Trajectories that eventually hit a power of 2

But just like the 3 x + 1 problem, the
 3 x  −  1
problem also has trajectories that eventually hit a power of 2 (thus falling straight down to 1, like hailstones). The easiest examples to find (besides the obvious) are numbers of the form
${\displaystyle {\frac {2^{2n+1}+1}{3}}\cdot 2^{k},\quad n\geq 0,\ k\geq 0,\,}$
which always hit
 2 2n +1
.

### Table of trajectories

 3 x  −  1
problem” trajectories

 n
A-number Trajectory (until known) Comment
1 {1, 2, 1, ...} Trivial cycle {1, ..., 1} with length 2
3 {3, 8, 4, 2, 1, ...} Hits 2 3
5 A003079 {5, 14, 7, 20, 10, 5, ...} Cyclic {5, ..., 5} with length 5
6 {6, 3, ...} Hits 2 3
9 {9, 26, 13, 38, 19, 56, 28, 14, ...} Eventually cyclic {14, ..., 14} with length 5
11 {11, 32, 16, 8, ...} Hits 2 5
12 {12, 6, ...} Hits 2 3
15 {15, 44, 22, 11, ...} Hits 2 5
17 A003124 {17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17, ...} Cyclic {17, ..., 17} with length 18
18 {18, 9, ...} Eventually cyclic {14, ..., 14} with length 5
21 {21, 62, 31, 92, 46, 23, 68, ...} Eventually cyclic {68, ..., 68} with length 18
24 {24, 12, ...} Hits 2 3
27 {27, 80, 40, 20, ...} Eventually cyclic {20, ..., 20} with length 5
29 {29, 86, 43, 128, 64, 32, ...} Hits 2 7
30 {30, 15, ...} Hits 2 5
33 {33, 98, 49, 146, 73, 218, 109, 326, 163, 488, 244, 122, ...} Eventually cyclic {122, ..., 122} with length 18
35 {35, 104, 52, 26, ...} Eventually cyclic {14, ..., 14} with length 5
36 A008894 {36, 18, ...} Eventually cyclic {14, ..., 14} with length 5