Highly composite number

A positive integer with more positive divisors than every smaller positive integer. Its prime factorization has nonincreasing exponents across increasing primes.

Divisor function: The function that counts the positive divisors of an integer. Highly composite numbers are precisely the integers that establish successive records for this function.
Srinivasa Ramanujan: An Indian mathematician whose work transformed number theory in the early twentieth century. His 1915 paper introduced the term and systematically studied these integers.
Least common multiple: The smallest positive integer divisible by each of a given set of integers. Highly composite numbers often arise as least common multiples of consecutive integers.
Superior highly composite number: An integer that maximizes the ratio of its divisor count to a positive power of the integer for some exponent. These analytically defined relatives include every highly composite number but impose a stronger optimization condition.
1: The multiplicative identity and the only positive integer with no prime factors. It is the first highly composite number, with one positive divisor.
Prime factorization: The expression of an integer as a product of prime powers. The exponents in this expression determine how many divisors the integer has.
Highly Composite Numbers (Ramanujan paper): A 1915 paper by Srinivasa Ramanujan on integers with record numbers of divisors. It established the foundational theory and named the class.
Highly divisible number: An integer with more divisors than every integer below it under a specified comparison range or rule. This broader description distinguishes the global record property from related local notions.
Colossally abundant number: An integer that maximizes the divisor-count ratio against a power of its size for some positive exponent. This related record sequence overlaps substantially but does not coincide with highly composite numbers.
2: The smallest prime number and the only even prime. It is the next highly composite number, with two positive divisors.

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