Superior highly composite number

In mathematics, a superior highly composite number is a natural number for which there is an ε > 0 such that for all natural numbers k ≥ 1,

$\frac{d(n)}{n^\varepsilon}\geq\frac{d(k)}{k^\varepsilon}$

where d(n), the divisor function, denotes the number of divisors of n. The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200... (sequence A002201 in OEIS).

Properties

All superior highly composite numbers are highly composite; it can also be shown that there exist prime numbers π₁, π₂, ... such that the n-th superior highly composite number s_n can be written as

$s_n = \prod_{i=1}^n\pi_i$

The first few π_n are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in OEIS). In other words, the ratio between two successive superior highly composite numbers is equal to a prime number.

References

• Srinivasa Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. 14, 347-407, 1915; reprinted in Collected Papers (Ed. G. H. Hardy et al.), New York: Chelsea, pp. 78–129, 1962
• Weisstein, Eric W., "Superior highly composite number" from MathWorld.