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Encyclopedia > 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, Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

frac{d(n)}{n^varepsilon}geqfrac{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). In mathematics the divisor function σa(n) is defined as the sum of the ath powers of the divisors of n, or The notations d(n) and (the tau function) are also used to denote σ0(n), or the number of divisors of n. ... 2 (two) is a number, numeral, and glyph. ... 6 (six) is the natural number following 5 and preceding 7. ... 12 (twelve) is the natural number following 11 and preceding 13. ... 60 (sixty) is the natural number following 59 and preceding 61. ... 120 (one hundred twenty in American English; one hundred and twenty in British English) is the natural number following 119 and preceding 121. ... 360（Three hundred sixty） is the natural number following 359 and preceding 361. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

Contents


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 A highly composite number is a positive integer which has more divisors than any positive integer below it. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

s_n = prod_{i=1}^npi_i

The first few πn are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in OEIS). The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...


See also

A composite number is a positive integer which has a positive divisor other than one or itself. ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ...

External links

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

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PlanetMath: highly composite number (84 words)
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The history of composite materials has developed from humble beginnings such as pre-history mud bricks reinforced with straw, Mesopotamian boats which were made from reed bundles reinforced with bitumen, American Indian wood and mud structures, etc., to a major topic of present-day materials science.
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