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Highly Composite Number Checker

Enter a positive integer to test whether it is a highly composite number.

How it is calculated

The tool counts the divisors of n by testing every i up to √n: each divisor i pairs with n ÷ i, so a pair adds 2 to the count and a perfect square root adds only 1. Whether n is highly composite is then read from the known list of highly composite numbers up to the 2,000,000 limit (1, 2, 4, 6, 12, 24, … 720720, 1081080, 1441440), so the answer is instant. By definition such a number has strictly more divisors than every smaller k; a tie with any smaller number disqualifies it. The result shows Yes or No, plus the divisor count. A decimal input is rounded to the nearest integer, and values below 1 or above 2,000,000 give no result.

Worked example

Check n = 120. Its divisors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120, which is 16 divisors. The largest count among smaller numbers is 12, reached by 60, so none of them reaches 16 and 120 is highly composite (result 1). By contrast 72 also has 12 divisors, but 60 already has 12, so the tie means 72 is not highly composite (result 0).

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Common mistakes

  • Assuming a number with many divisors must qualify: 72 has 12 divisors, yet it is not highly composite because the smaller number 60 has 12 as well.
  • Confusing highly composite with merely composite: every composite number has more than two divisors, but only a record-setting few are highly composite.
  • Counting 1 and the number itself wrongly when listing divisors by hand: both count, so 12 has 6 divisors, not 4.
  • Typing a decimal such as 59.6 and expecting an error: the input is rounded to 60 first, so the answer is for 60.

About this calculator

A highly composite number is a positive integer that has more divisors than any smaller positive integer — a concept formally studied by Srinivasa Ramanujan in a landmark 1915 paper. For example, 12 is highly composite because it has 6 divisors (1, 2, 3, 4, 6, 12), more than any integer below it, while 10 is not, since 8 already has 4 divisors, matching 10's own count without beating it. The sequence starts 1, 2, 4, 6, 12, 24, 36, 48, 60, 120... and grows increasingly sparse.

These numbers are of interest in number theory research, recreational mathematics, and combinatorics, and they show up practically wherever a number with many factors is useful — historically, 12, 24, 60, and 360 (all highly composite or closely related) were chosen for hours in a day, minutes in an hour, and degrees in a circle precisely because they divide evenly by so many small numbers.

For any positive integer up to 2,000,000 it counts the divisors and checks the number against the known list of highly composite numbers, useful for number-theory students and enthusiasts checking Ramanujan's list or the OEIS sequence.

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Comments (1)

  • Editorial team

    A quick pre-screen before you run a large number: every highly composite number is built from consecutive primes starting at 2, with exponents that never increase. So 120 = 2³ × 3 × 5 passes the screen, while 90 = 2 × 3² × 5 fails it and cannot qualify. Passing is not enough, though: 72 = 2³ × 3² has the right shape and is still not highly composite. Which number made you curious enough to check it?

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