Commas, spaces, or underscores may group digits in threes. Decimals, zero, and negative integers are not accepted.
Try 60 to find the first nine highly composite numbers and see why 60 holds the record with 12 positive divisors.
A positive integer n is highly composite when its positive-divisor count is strictly larger than that of every smaller positive integer:
These are record holders, not merely numbers with many factors. For example, τ(12) = 6, which beats every earlier count, so 12 is highly composite. The number 18 also has 6 divisors, but it only ties 12 and does not set a new record.
If n = p1a₁p2a₂⋯praᵣ, then:
A record holder uses consecutive primes starting with 2, and its exponents can be written in nonincreasing order: a₁ ≥ a₂ ≥ ⋯ ≥ aᵣ. Otherwise, moving a larger exponent to a smaller prime would produce a smaller integer with the same divisor count.
The finder recursively builds those ordered prime-exponent combinations up to the limit, sorts them by value, and retains a candidate only when its exact τ value exceeds every earlier maximum.
This method uses exact BigInt arithmetic and a background worker where supported. It is much smaller than scanning all integers through the limit, while still covering every possible record holder in the supported range.
The prime factorization is 60 = 22 × 3 × 5. Each divisor independently chooses an exponent of 0, 1, or 2 for the prime 2, and an exponent of 0 or 1 for both 3 and 5.
The preceding highly composite number is 48, which has 10 divisors. No positive integer below 60 has 12 divisors, so 60 sets a new record.
It is a positive integer with strictly more positive divisors than every smaller positive integer. The first values are 1, 2, 4, 6, 12, 24, 36, 48, and 60.
Yes under the standard record-setting definition used by OEIS A002182 and this finder. It starts the sequence with one divisor, and no smaller positive integer exists.
No. A composite number merely has more than two positive divisors. A highly composite number must beat every earlier divisor count. Also, 1 and 2 start the highly composite sequence even though neither is composite.
No. The inequality is strict. For instance, both 12 and 18 have six positive divisors, but only 12 is the first to reach six and therefore the record holder.
For a fixed exponent pattern, assigning larger exponents to smaller primes minimizes the number without changing its divisor count. A smaller number reaches that divisor count sooner, which is necessary for setting a record.
Yes. “Anti-prime” is an informal name often used for highly composite numbers. It does not mean the number is the opposite of a prime in a formal algebraic sense.
The accepted upper limit is 1 quintillion (1018). Calculations are exact; the limit is chosen to keep generation and rendering practical on typical browsers and mobile devices.
Yes. Candidate generation, sorting, and record comparison happen locally in your browser. The finder does not upload, store, or add your number to the page URL.
