Find a pair that is similar to $2401:343$.

Let's understand the mathematical relationship in the given base pair $2401:343$:
* Both numbers are powers of $7$.
* $2401=7^4$
* $343=7^3$
So the relationship pattern is $x^4:x^3$.

Let's test the options to find a matching pattern:

* Option A: $256:64$

$$256=4^4$$

$$64=4^3$$

This matches the pattern $x^4:x^3$ exactly for $x=4$.
* Option B: $32:4⟹2^5:2^2$ (Does not match)
* Option C: $216:1916⟹6^3:1916$ (Does not match)
* Option D: $9:27⟹3^2:3^3$ (Does not match)

Thus, the correct option is A.