Highly composite numbers

A highly composite number (or anti-prime) is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.

The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer.

The initial or smallest 38 highly composite numbers are listed in the table below (sequence in the OEIS). The number of divisors is given in the column labeled d(n).

The table below shows all the divisors of one of these numbers.

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:

where ${\displaystyle a_{n}}$ is the sequence of successive prime numbers, and all omitted terms (a₂₂ to a₂₂₈) are factors with exponent equal to one (i.e. the number is ${\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451}$).

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