highly composite number

A highly composite number is a positive integer which has more divisors than any positive integer below it. The first twenty highly composite numbers are

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560 and 10080.

This sequence is Sloane A002182, with 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64 and 72 positive divisors, respectively (A002183). The sequence of highly composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors, A005179. There are an infinite number of highly composite numbers. To prove this fact, suppose that n is an arbitrary highly composite number. Then 2n has more divisors than n (2n is a divisor and so are all the divisors of n) and so some number larger than n (and not larger than 2n) must be highly composite as well. Roughly speaking, for a number to be a highly composite it has to have prime factors as small as possible, but not too many of the same. If we decompose a number n in prime factors like this:

n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}

where

p_1 < p_2 < \cdots < p_k

are prime, and the exponents

c_i

are positive integers, then the number of divisors of n is exactly

(c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1)

Hence, for n to be a highly composite number,

the k given prime numbers $p_i$ must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have 4 divisors);
the sequence of exponents must be non-increasing, that is $c_1 \geq c_2 \geq \cdots \geq c_k$ ; otherwise, by exchanging two faulty exponents we would again get a smaller number than n with the same number of divisors (for instance 18=2¹x3² may be replaced with 12=2²x3¹, both have 6 divisors).

Also, except in two special cases n = 4 and n = 36, the last exponent c_k must equate 1. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. Highly composite numbers higher than 6 are also abundant numbers. One need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. Many of these numbers are used in traditional systems of measurement, and tend to be used in engineering designs, due to their ease of use in calculations involving fractions. If Q(x) denotes the number of highly composite numbers which are less than or equal to x, then there exist two constants a and b, both bigger than 1, so that

(ln x)^a ≤ Q(x) ≤ (ln x)^b.

with the first part of the inequality proved by Paul Erdős in 1944 and the second part by J.-L. Nicholas in 1988.

External link

MathWorld: Highly Composite Number

<< Previous

Word Browser

Next >>

grand canne
east turkestan islamic movement
selim iii
mahmud ii
mustafa iv
boyfriend (fashion)
fulmar
promulgation
virgin islands march
x bar theory
george balanchine

non standard cosmology
piaggio aero
tropicbird
mehmed v
bell x 1
verb phrase
jean baptiste camille corot
social credit
astronomical journal
new zealand flax
gulbene

spaceguard
gudea
list of songs about bipolar disorder
paracimexomys group
stella rimington
paracimexomys
west riding of yorkshire
mimas (moon)
arcsec
bryceomys
james glaisher

tubeway army
waldo semon
jess lidyard
m
western finland
lovedean
southern finland
eastern finland
shotokan
world values survey
ferdinando fairfax, 2nd baron fairfax of cameron

Highly Composite Number

See also

External link