Highly powerful number

In elementary number theory, a highly powerful number is a positive integer that satisfies a property introduced by the Indo-Canadian mathematician Mathukumalli V. Subbarao.^[1] The set of highly powerful numbers is a proper subset of the set of powerful numbers.

Define prodex(1) = 1. Let ${\displaystyle n}$ be a positive integer, such that ${\displaystyle n=\prod _{i=1}^{k}p_{i}^{e_{p_{i}}(n)}}$, where ${\displaystyle p_{1},\ldots ,p_{k}}$ are ${\displaystyle k}$ distinct primes in increasing order and ${\displaystyle e_{p_{i}}(n)}$ is a positive integer for ${\displaystyle i=1,\ldots ,k}$. Define ${\displaystyle \operatorname {prodex} (n)=\prod _{i=1}^{k}e_{p_{i}}(n)}$. (sequence A005361 in the OEIS) The positive integer ${\displaystyle n}$ is defined to be a highly powerful number if and only if, for every positive integer ${\displaystyle m,\,1\leq m<n}$ implies that ${\displaystyle \operatorname {prodex} (m)<\operatorname {prodex} (n).}$^[2]

The first 25 highly powerful numbers are: 1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400. (sequence A005934 in the OEIS)