For Hailstone numbers, what is the maximum number of iterations to reach the cycle (4, 2, 1....) and which number produces this max?

Hailstone Numbers

I'm a little slow to answer this, because I didn't know the sequence as "Hailstone numbers"; I learned about them from Douglas Hofstadter's column in which he referred to the function as the

"half or triple plus one" function.

[He then abbreviated it hotpo(), and named its argument "tato," standing for "tato and tato ONLY." This allowed him to write code in the form

hotpo(tato)

of course!]

Anyhow, if you think a bit, I believe you'll recognize that there IS no maximum number of iterations, because for any number N that would produce such a suspected maximum, there is the sequence

N, 2N, 4N, 8N, ...

each of which produces one more iteration than the one which precedes it in the sequence.