Superior highly composite number

In mathematics, a superior highly composite number is a certain kind of natural number. Formally, a natural number n is called superior highly composite iff there is an ε > 0 such that for all natural numbers k ≥ 1,



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 sn can be written as



The first few πn are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in OEIS).

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

* Eric W. Weisstein, Superior highly composite number at MathWorld.

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