Superior highly composite number

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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,

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

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

1 Properties
2 See also
- 2.1 External links
3 References

[edit] 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).

[edit] See also

[edit] External links

MathWorld: Superior highly composite number

[edit] 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