M/n sequences

The m/n sequences generalize the well-known 3x+1 problem, which is nothing else than the 3/2 sequence in this terminology. These sequences are orbits of the map

$x\mapsto {\begin{cases}\lfloor {\frac {m\,x}{n}}\rfloor +1&x{\rm {~~odd}}\\~~~x/2&x{\rm {~~even}}\end{cases}}.$ .

These sequences have been mentioned by Yasutoshi Kohmoto on the SeqFan list in September 2009, but have certainly been studied earlier.^{[Please provide a reference.]}

It appears that for some values, all orbits are (eventually) periodic, while for other values of m/n, some orbits tend to infinity. (There is obviously no other possibility.)

Examples

Collatz sequence

The 3x+1 problem corresponds to the m/n = 3/2 sequence, it is discussed in much detail in the main entry for the 3x+1 problem.

8/5 sequence

The map for the 8/5 sequence is OEIS sequence A082010.

The orbit of 7 under this map is sequence A152199. I includes the orbit of 3, 5, 6, 7, 9, ... as subsequence.

The orbit of 1 (and 2) is the periodic sequence A000034 = (1, 2, 1, 2, 1, 2,...).

The next "missing" number is 11 with orbit (11, 18, 9, 15, 25, 41, 66, 33, 53, 85, 137, 220, 110, 55, 89, 143, 229, 367, 588, 294, 147, ...). However, the orbit of 13 contains this one as a subsequence.

See also

• 3x+1 problem (a.k.a. Collatz problem)