Large highly powerful numbers are cubeful

A number $n = \prod\nolimits_{i = 1}^k {p_i^E}$ is called highly powerful if the product of the exponents $E({p_i})$ of the primes is larger than that of any smaller number. If ${p_k} > 19$, $E({p_k}) = 3$. Further, we have developed an algorithm which finds all highly powerful numbers with $E({p_k}) \ne 3$, and we list the 19 highly powerful numbers with $E({p_k}) = 2$.