Digit Sums of Exponentiated Numbers

Number Theory Level 5

\[\large 8^3=512\Rightarrow5+1+2=8\]

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 \(n=1\).

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