Highly Composite Numbers (HCN)

Doing Math with JavaScript

Input an integer n (0 < n < 1020)		Result: is n highly composite?1     Divisors   Prime Factors	Run time:

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.
For example, n = 120 is highly composite because it has 16 divisors (so d(120) = 16) and all smaller integers have fewer than 16 divisors.

The concept of HCN was introduced by Srinivasa Ramanujan in 1915. Here are excerpts from his paper
(Proceedings of the London Mathematical Society, 2, xiv, 1915, 347-409):
• Highly Composite Numbers – title page of the paper
• Ramanujan's definition of highly composite numbers
• List of highly composite numbers, page 1 (note: Ramanujan did not put 1 on the HCN list!)
• List of highly composite numbers, page 2 (with HCN 293318625600 omitted).

Input a number n (up to 20 digits), and this calculator will check if n is highly composite. Enjoy!

See also:
• 100+ digit calculator: arbitrary precision arithmetic
• Prime factorization calculator
• Euler's totient function φ(n)
• Divisors and sum-of-divisors calculator
• Binomial Coefficients Calculator
• Fibonacci numbers calculator
• Catalan numbers calculator

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