Highly powerful number

Last updated April 03, 2024

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 $n=\prod _{i=1}^{k}p_{i}^{e_{p_{i}}(n)}$ , 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 $\operatorname {prodex} (n)=\prod _{i=1}^{k}e_{p_{i}}(n)$ . (sequence A005361 in the OEIS ) The positive integer $n$ is defined to be a highly powerful number if and only if, for every positive integer $m,\,1\leq m<n$ 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. (sequence A005934 in the OEIS )

Related Research Articles

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer $n$ ,

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a, 0) as 0 for all a, since 0 is the only common multiple of a and 0.

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.

In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 3². The smallest positive square-free numbers are

In number theory, given a prime number $p$ , the $p$ -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; $p$ -adic numbers can be written in a form similar to decimals, but with digits based on a prime number $p$ rather than ten, and extending to the left rather than to the right.

2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.

63 (sixty-three) is the natural number following 62 and preceding 64.

In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

A powerful number is a positive integer m such that for every prime number p dividing m, p² also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a²b³, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x₁, ..., x_N has a low discrepancy.

A highly totient number $is an integer that has more solutions to the equation, where is Euler's totient function, than any integer smaller than it. The first few highly totient numbers are$

<span class="mw-page-title-main">Carmichael function</span> Function in mathematical number theory

In number theory, a branch of mathematics, the Carmichael function $λ (n)$ of a positive integer $n$ is the smallest member of the set of positive integers $m$ having the property that

In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product:

In mathematics, a superabundant number is a certain kind of natural number. A natural number $n$ is called superabundant precisely when, for all $m < n$ :

In number theory, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.

<span class="mw-page-title-main">Superior highly composite number</span> Class of natural numbers

In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.

288 is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".

In mathematics, the Fibonacci numbers form a sequence defined recursively by:

In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.

In number theory, Jordan's totient function, denoted as $, where is a positive integer, is a function of a positive integer,, that equals the number of -tuples of positive integers that are less than or equal to and that together with form a coprime set of integers$

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 .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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 .

[1]

[2]