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.{{cite book|author=Hardy, G. E.|author2=Subbarao, M. V.|chapter=Highly powerful numbers|title=Congr. Numer. 37|year=1983|pages=277–307}} 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 $p\_1,\; \backslash ldots\; ,\; p\_k$ are $k$ distinct primes in increasing order and $e\_\{p\_i\}(n)$ is a positive integer for $i\; =\; 1,\; \backslash ldots\; ,k$. Define $\backslash operatorname\{prodex\}(n)\; =\; \backslash prod\_\{i=1\}^k\; e\_\{p\_i\}(n)$. {{OEIS|id=A005361}} The positive integer $n$ is defined to be a highly powerful number if and only if, for every positive integer implies that {{cite journal|title=Large highly powerful numbers are cubeful|author=Lacampagne, C. B.|authorlink=Carole Lacampagne|author2=Selfridge, J. L.|authorlink2=John Selfridge|journal=Proceedings of the American Mathematical Society|volume=91|issue=2|date=June 1984|pages=173–181|doi=10.1090/s0002-9939-1984-0740165-6|doi-access=free}}

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. {{OEIS|id=A005934}}