I have read in a book that has very much of a recreational flavour that if we take any natural number and square its digits and add that then if we repeat that long enough that only two outcomes can happen:

a) Either we stay forever in a trivial cycle $1$

or

b) We stay forever in a cycle $145, 42, 20, 4, 16, 37, 58, 89$.

But I do not now how a general situation looks like.

Do we have a formula that tells us how many exactly cycles do we have if we take $k$-th power, that is, how to calculate number of possible cycles as a function of $k$?

If we denote that function by $C(k)$ is it known is $C$ a bounded function? (In other words, is a set $\{C(1),C(2),...C(k),...\}$ bounded?)