Prove that there exists $k>1$ and $x_1, x_2, \dots ,x_k$ such that $\sum_{i=1}^k\phi(x_i)$ and $\sum_{i=1}^kx_i$ are both $n$th powers of an integer I got an interesting question which asks

Consider a fixed positive integer $n>1$. Prove that there exists an
integer $k>1$ and distinct integers $x_1, x_2, \dots ,x_k$, all
greater than $1$, such that $\sum_{i=1}^k\phi(x_i)$ and $\sum_{i=1}^kx_i$ are both $n$th powers of an integer. Here $\phi$ is the Euler
totient function.

I am familiar with the basic elementary properties of the $\phi$ function, and I tried to write down the elaborate formula of $\phi(n)$ and manipulate the two sums to arrive at a conclusion. But that didn't help. Things became too ugly too soon. I also tried to look at $\sum_{i=1}^k(\phi(x_i)-x_i)$ but couldn't proceed anywhere.
Any help would be appreciated.
 A: Select a provisional set of values $y_1=3,y_2=5,\dots,y_k=2k+1$ and let the sum of those values of $\varphi(y_i)$ be equal to $S$. If $S$ is not perfect $n$'th power let $m$ be the difference to the next perfect $n$'th power and write $m$ as a sum of $d$ powers of $2:$ $2^{a_1},2^{a_2},\dots,2^{a_d}$ and let $y_{k+i} = 2^{a_i+1}$. Note that $m<k^2$ for large values of $k$.
At this point we have $k+d$ distinct values such that the sum of the values of $\varphi(y_i)$ is a perfect $n$'th power.
What we are going to do to make the sum of the $x_i$ also a perfect $n$'th power is select some of the initial $k$ values $x_i$ to be $y_i$ and some $2y_i$, and let $x_{k+i} = y_{k+i}$ for $i>0$. So we are going to be taking $k+d$ values $x_i$.
By doing this the sum of the values of $\varphi(x_i)$ is equal to the sum of values of $\varphi(y_i)$ and hence is a perfect $n$'th power, and the sum of the values of $x_i$ can be equal to $s+ \sum\limits_{i=1}^{k+d}y_i$ where $s$ is the sum of any subset of $\{3,5,\dots,2k+1\}$
We can show that for $k\geq 10$ every $s$ in the range $[8,k^2/10]$ is possible by induction.
Since we have that $t=\sum\limits_{i=1}^{k+d}y_i$ is at most $2k^2$ for large $k$, it follows that for large $k$ there is a perfect $n$'th power in the range $[t+8,t+k^2/10]$.
