Highly powerful number explained

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 n = \prod ^{k} _{i =1} p ^{e _{p _{i}} (n)} _{i} , where p _{1,} \ldots , p _{k} are k distinct primes in increasing order and e _{p _{i}} (n) is a positive integer for i = 1 , \ldots ,k . Define \operatorname{prodex}(n) = \prod ^{k} _{i =1} e _{p _{i}} (n) . 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 \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.

Notes and References

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