Numbers, exponents and cycles

Suppose we have a number . We apply the following procedure to in order to obtain a new number : we take ‘s digits and raise them alternatively to 2 and 4, summing the results. Thus, will become . We don’t stop here. Instead, we apply the same procedure to obtaining .

Of course, and someone can prove it, this procedure will quickly lead either to a number already seen or to 1. For now, we are only interested in the number of iterations required for this.