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The
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
always reaches
1 when
, with the
Collatz function defined as
 $f(n)={\begin{cases}n/2&{\mbox{if }}n\equiv 0{\pmod {2}},\\3n+1&{\mbox{if }}n\equiv 1{\pmod {2}}.\end{cases}}\,$
The
problem instead uses the
iterated function
 $f(n)={\begin{cases}n/2&{\mbox{if }}n\equiv 0{\pmod {2}},\\3n1&{\mbox{if }}n\equiv 1{\pmod {2}}.\end{cases}}\,$
The 3 x − 1 problem trajectories
The
problem trajectories might be of 3 kinds
 Trajectories heading towards infinity;
 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, ...).
Trajectories heading towards infinity
Do
problem trajectories heading towards infinity exist or not?
Eventually nontrivially cyclic trajectories
The
problem does have eventually nontrivially cyclic trajectories!
A003124 One of the basic cycles (of length 18) in the
(
odd) or
(
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
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
 ${\frac {2^{2n+1}+1}{3}}\cdot 2^{k},\quad n\geq 0,\ k\geq 0,\,$
which always hit
.
Table of trajectories
“ problem” trajectories

Anumber

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

See also