structural:scholar_edition
Empirical data

For instance, starting with n = 12 and applying the function f without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.

The number n = 19 takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1.

Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with:

The starting values whose maximum trajectory point is greater than that of any smaller starting value are as follows:

Number of steps for n to reach 1 are

The starting value having the largest total stopping time while being

These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, 9 780 657 631 has 1132 steps, as does 9 780 657 630.

The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the powers of two since 2 is halved n times to reach 1, and is never increased.