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Highly powerful number

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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 {\displaystyle n} be a positive integer, such that n = ∏ i = 1 k p i e p i ( n ) {\displaystyle n=\prod _{i=1}^{k}p_{i}^{e_{p_{i}}(n)}} , where p 1 , … , p k {\displaystyle p_{1},\ldots ,p_{k}} are k {\displaystyle k} distinct primes in increasing order and e p i ( n ) {\displaystyle e_{p_{i}}(n)} is a positive integer for i = 1 , … , k {\displaystyle i=1,\ldots ,k} . Define prodex ⁡ ( n ) = ∏ i = 1 k e p i ( n ) {\displaystyle \operatorname {prodex} (n)=\prod _{i=1}^{k}e_{p_{i}}(n)} . (sequence A005361 in the OEIS) The positive integer n {\displaystyle n} is defined to be a highly powerful number if and only if, for every positive integer m , 1 ≤ m < n {\displaystyle m,\,1\leq m<n} implies that prodex ⁡ ( m ) < prodex ⁡ ( n ) . {\displaystyle \operatorname {prodex} (m)<\operatorname {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)

References

  1. Hardy, G. E.; Subbarao, M. V. (1983). "Highly powerful numbers". Congr. Numer. 37. pp. 277–307.
  2. 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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