# Highly composite number

In mathematics, a superior highly composite number is a certain kind of natural number with the following properties a natural number n is called superior highly composite iff there is an ε 0 such that for all natural numbers k ≥ 1, where d(n), the divisor function, denotes the number of divisors of nthe first few superior highly composite numbers. An algorithm for generating highly composite numbers d b siano and j d siano oct 7, 1994 introduction all positive integers can be written in only one way as a. Highly composite number highly composite numbers are numbers such that divisor function (ie, the number of divisors of ) is greater than for any smaller superabundant numbers are closely related to highly composite numbers, and the first 19 superabundant and highly composite numbers are the same. Volume 72, number 241, pages 387{395 s 0025-5718(02)01419-9 article electronically published on march 5, 2002 rapid multiplication modulo the sum and difference of highly composite numbers colin percival abstract we extend the work of richard crandall et al to demonstrate how the discrete weighted transform (dwt) can be applied to speed up multiplication modulo any number. [ today's exercise was written by zacharias voulgaris, based on a numberphile video guest authors are always welcome contact me if you wish to write an exercise ] a highly composite number, also called an anti-prime, is a number n for which d(m) d(n) for all m n, where d(x) is the divisor.

I thought i would wrote a small article on some numbers which i find interesting, i may expand upon this topic in the future, but for this article i'm going to restrict myself to three forms of number: kaprekar's constant, polygonal numbers and highly composite numbers. The highly composite number 60, for example, has become our standard for measuring time as it can be split into convenient equal portions of whole numbers—30 minutes (half an hour), 15 minutes (a quarter of an hour), 10 minutes (a sixth of an hour), and so on 360 is likewise a highly composite number, and the foundation of circular geometry. S ramanujan, highly composite numbers, annotated and with a foreword by j-l nicolas and g robin, ramanujan j, 1 (1997), 119-153 s ramanujan, highly composite numbers: section iv, in 1) collected papers of srinivasa ramanujan, pp 111-8, ed g h hardy et al, ams chelsea 2000 2) ramanujan's papers, pp 143-150, ed b j.

In 1915, the london mathematical society published in its proceedings a paper by ramanujan entitled highly composite numbers a number n is said to be highly composite if for every integer mn, it happens that. English: plot of the number of divisors of integers from 1 to 1000, colour-coded by their units digit highly-composite numbers are labelled in bold and superior highly-composite numbers are starred prime numbers are those with exactly 2 divisors. As the others have said, 8 is the first factorial that is not a highly composite number (see oeis (link)) 9 is also not highly composite the number of different positive divisors n (n≥2) has is.

How to identify if a given number is a composite number or not i'd go with ignacio's solution but if you wanted to make it more efficient (with respect to time, at the cost of a little space), you can just divide by all the primes between 2 and the square root of n that's because, if a number is not divisible by 2, there's no point checking if it's. Judah asks: a highly composite number is a number that has more factors than any other less than it (positive, whole numbers) like 12 has 6 factors, which is more than any other below it (whereas 1 has 1 , 2 has 2, and so on.

On the opposite extreme from a prime number is a highly composite number along with a composite number, a highly composite number is one of two types of number that is the opposite of a prime numbera composite number is the general opposite to a prime a highly composite number (hcn) is the special opposite of a prime the set of hcns is a subset of composite numbers. Find whether natural number is composite or prime akimbas akimbasas ranch hand posts: 32 1 posted 4 years ago hello, i need to write a program that finds out whether the number is prime or composite, if the number is composite, the program should write what are the divisors and how many are there if the number is prime, it need to say that he number. I thought i would wrote a small article on some numbers which i find interesting, i may expand upon this topic in the future, but for this article i'm going to restrict myself to three forms of number: kaprekar's constant, polygonal numbers and highly composite numbers kaprekar's constant: kaprekar's constant is a special. A positive integer n is called a highly composite number if it has more divisors than any predecessor let d(n) be the number of divisors of n, then n is highly composite if d(m) d(n) for all positive m n.

## Highly composite number

A composite number (also called a superabundant number) is a number which has more factors than any other number less than in other words, exceeds for all , where is the divisor functionthey were called highly composite numbers by ramanujan, who found the first 100 or so, and superabundant by alaoglu and erdös (1944) there are an infinite number of highly composite numbers. How does the number tree grow as the numbers descend from the heavens and attach themselves to the growing tree, they look for a specific number on a branch point.

• Find all divisors of the input number n, the total number of divisors d(n), and the sum of divisors the input n can be up to 20 digits.
• For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have and for all natural numbers k larger than n we have where d(n), the divisor function, denotes the number of divisors of n the term was coined by ramanujan (1915.
• A superior highly composite number is a positive integer for which there is an such that for all , where the function counts the divisors of (ramanujan 1962, pp 87 and 115) it.

On the opposite extreme from a prime number is a highly composite numbera prime has the least quantity of factors, namely two a highly composite number is one which has a greater number of factors than any smaller number. I'd like to build an efficient python iterator/generator that yields: all composite numbers less than n along with their prime factorization i'll call it. - today i was thinking about what the opposite of a prime number would be, and hcns turned out to be my most. Imagine to divide the members of this family by a number n and compute the remainders should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the casesthis outcome is represented by a white square.