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The sequence of highly composite numbers (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS ). Highly composite numbers whose number of divisors is also a highly composite number are. 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600 ...
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. For any possible exponent, whichever integer has the greatest ratio is a superior ...
A number n that has more divisors than any x < n is a highly composite number (though the first two such numbers are 1 and 2). Composite numbers have also been called "rectangular numbers", but that name can also refer to the pronic numbers , numbers that are the product of two consecutive integers.
In mathematics. 360 is a highly composite number [1] and one of only seven numbers such that no number less than twice as much has more divisors; the others are 1, 2, 6, 12, 60, and 2520 (sequence A072938 in the OEIS ). 360 is also a superior highly composite number, a colossally abundant number, a refactorable number, a 5- smooth number, and a ...
Then in particular any superabundant number is an even integer, and it is a multiple of the k-th primorial #. In fact, the last exponent a k is equal to 1 except when n is 4 or 36. Superabundant numbers are closely related to highly composite numbers. Not all superabundant numbers are highly composite numbers.
Table of divisors. Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics. The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n ...
Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on highly composite numbers. [2] Unfortunately, the publisher of the journal to which Ramanujan submitted his work, the London Mathematical Society, was in financial difficulties at the time and Ramanujan agreed to remove aspects of the work to reduce the cost of printing ...
In number theory, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by Pillai ( 1943 ), and early work on the subject was done by ...