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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 n be a positive integer, such that \( {\displaystyle n=\prod _{i=1}^{k}p_{i}^{E(p_{i})}} \), where \( {\displaystyle p_{1},\ldots ,p_{k}} \) are k distinct primes in increasing order and \( {\displaystyle E(p_{i})} \) is a positive integer for \( {\displaystyle i=1,\ldots ,k} \). Define \( {\displaystyle \operatorname {prodex} (n)=\prod _{i=1}^{k}E(p_{i})} \) The positive integer 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)
References

Hardy, G. E.; Subbarao, M. V. (1983). "Highly powerful numbers". Congr. Numer. 37. pp. 277–307.
Lacampagne, C. B.; Selfridge, J. L. (June 1984). "Large highly powerful numbers are cubeful". Proceedings of the American Mathematical Society. 91 (2): 173–181. doi:10.1090/s0002-9939-1984-0740165-6.

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