Consecutive prime powers that are not prime

Question

I am looking for consecutive entries $(a_k,a_{k+1})$ in the sequence of prime powers $$(a_n)=(2,3,2^2,5,7,2^3,3^2,11,13,2^4,17,19,23,5^2,3^3,29,31,2^5\cdots)$$ such that neither $a_k$ nor $a_{k+1}$ are primes. Examples are $(2^3,3^2), (5^2,3^3), (3^7,13^3)$. I realize this may be a very difficult problem, but I wonder how many have been found and what is known.