Highly powerful number

Short description: Positive integers that have a property about their divisors

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. The set of highly powerful numbers is a proper subset of the set of powerful numbers.

Define prodex(1) = 1. Let $n$ be a positive integer, such that $n=\prod _{i=1}^k{p}_i^{e_{p_i}(n)}$, where $p_1,\dots ,p_k$ are $k$ distinct primes in increasing order and $e_{p_i}(n)$ is a positive integer for $i=1,\dots ,k$. Define $\mathrm{prodex}(n)=\prod _{i=1}^k{e}_{p_i}(n)$. (sequence A005361 in the OEIS) The positive integer $n$ is defined to be a highly powerful number if and only if, for every positive integer $m,1\le m<n$ implies that $\mathrm{prodex}(m)<\mathrm{prodex}(n).$

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)