Superior highly composite number

From Wikipedia, the free encyclopedia

Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Sociable number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization

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), 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).

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

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

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

  • 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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