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. The set of highly powerful numbers is a proper subset of the set of powerful numbers. Define prodex(1) = 1. Let [math]\displaystyle{ n }[/math] be a positive integer, such that [math]\displaystyle{ n = \prod_{i=1}^k p_i^{e_{p_i}(n)} }[/math], where [math]\displaystyle{ p_1, \ldots , p_k }[/math] are [math]\displaystyle{ k }[/math] distinct primes in increasing order and [math]\displaystyle{ e_{p_i}(n) }[/math] is a positive integer for [math]\displaystyle{ i = 1, \ldots ,k }[/math]. Define [math]\displaystyle{ \operatorname{prodex}(n) = \prod_{i=1}^k e_{p_i}(n) }[/math]. (sequence A005361 in the OEIS) The positive integer [math]\displaystyle{ n }[/math] is defined to be a highly powerful number if and only if, for every positive integer [math]\displaystyle{ m,\, 1 \le m \lt n }[/math] implies that [math]\displaystyle{ \operatorname{prodex}(m) \lt \operatorname{prodex}(n). }[/math]

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)