Personal Investigation: Part Four

This post builds on my three previous posts so it's best to look at those first in order to properly understand what I'm doing. These posts are:

• Personal Investigation: Part 1
• Personal Investigation: Part 2
• Personal Investigation: Part 3

It occurred to me that there's probably no reason to maintain \(k\) at a constant value and that the exponents could be variable. For example:$$a_n= \text{ sum of digits } \big (a_{n-1}^2+a_{n-2}^3+a_{n-3}^4\big ) \text { with }a_0=0, a_1=1,a_2=2$$Do we eventually end up in a loop? This is what I set out to investigate.

Let's start with the following SageMath code (permalink):

a, b, c= 0, 1, 2
L=[a, b, c]
for x in [1..31]:
    d=a^2+b^3+c^4
    d=sum(d.digits())
    L.append(d)
    a, b, c=b, c, d
print(L)

Here, instead of a single value for \(k\), the values are 2,3 and 4. The output of the algorithm reveals that, very quickly, a loop arises:

1, 2, 8, 10, 13, 24, 20, 32, 21, 30, 25, 25, 17, 33, 30, 28, 28, 29, 18, 18, 22, 13, 23, 21, 24, 25, 25, 26, 24, 39, 28, 28, 29

It would seem that, regardless of the starting values or the exponents, the series always ends up looping. For example, take starting values of \(a, b, c\)= 100, 191, 227 and exponents of 112, 223 and 454. Thus:$$a_n= \text{ sum of digits } \big (a_{n-1}^{112}+a_{n-2}^{223}+a_{n-3}^{454}\big ) \text { with }a_0=100, a_1=191,a_2=227$$Here the series reaches a maximum value 8328 and contains 275 distinct terms.

I've only explored the triad here but there's no reason to suppose that similar results hold true for the tetrad, pentad, hexad and beyond. It would seem that for constant powers, no matter how high, the series eventually repeats.