# Digit Sums of Exponentiated Numbers

$\large 8^3=512\Rightarrow5+1+2=8$

Let us define the function $$S_k(n)$$ as the digit sum of the number $$n^k$$ where $$n,k\in\mathbb{N}$$. Shown above is an example of when $$S_k(n)=n$$. For each given $$k$$, there are a finite number of $$n$$ such that $$S_k(n)=n$$. In fact, sometimes for a given $$k$$ there are sets $$A=\{a_1,a_2,...a_n\}$$ such that the following property holds:

$S_k(a_1)=a_2, S_k(a_2)=a_3,\dots,S_k(a_n)=a_1$

An example of a set with this property would be $$A=\{13,16\}$$ for $$k=2$$.

$13^2=169\Rightarrow1+6+9=16 \\ 16^2=256\Rightarrow2+5+6=13$

Let us define $$U_k$$ as the multiset of all sets $$A$$ such that the above property holds for a given $$k$$. What is the minimum possible number of sets that $$U_k$$ could contain?

Details and Assumptions:

• Be sure to include the numbers $$n$$ where $$S_k(n)=n$$ into the calculation, these are just the special cases where $$S_k(a_n)=a_1$$ and $$n=1$$.
×