Here is another way to look at the problem. Assume the notation,
$$[a_1,a_2,\dots,a_m]^k = a_1^k + a_2^k +\dots +a_m^k$$
Theorem: (Tarry-Escott) If,
$$[a_1,a_2,\dots,a_m]^k = [b_1,b_2,\dots,b_m]^k,\;\; \text{for}\, k = 1,2,3,\dots,n$$
then for any constant $c$,
$$[a_1,\dots,a_m,\,c+b_1,\dots, c+b_m]^k = [b_1,\dots,b_m,\,c+a_1,\dots, c+a_m]^k \\ \text{for}\, k=1,2,3,\dots, n+1$$
This doubles the number of terms, but you can climb the ladder of powers as high as you like. For example, starting with,
$$[1,4]^k = [2,3]^k,\;\; \text{for}\, k = 1$$
using the theorem, we get,
$$[1,\,4,\,c_1+2,\,c_1+3]^k = [2,\,3,\,c_1+1,\,c_1+4]^k,\\ \text{for}\, k = 1,2$$
true for any $c_1$. If we use $c_1 = 4$, we recover your $[1,\,4,\,6,\,7]^k = [2,\,3,\,5,\,8]^k$. Using the theorem again, we have,
$$[1,\,4,\,6,\,7,\,c_2+2,\,c_2+3,\,c_2+5,\,c_2+8]^k = [2,\,3,\,5,\,8,\,c_2+1,\,c_2+4,\,c_2+6,\,c_2+7]^k,\\ \text{for}\, k = 1,2,3$$
true for any $c_2$, and where we can choose $c_2 = 8$ so terms are from $1,2,\dots, 16.$ And so on for any $k = 1,2,3,\dots,n$.