Highly made up Number

A highly made up number is a whole which has more Diviseur S that any positive entirety lower than him.

The blackjacks first numbers highly made up are:

There exists an infinity of highly made up numbers. For to prove this assertion, let us suppose that N is an arbitrary number highly compound. Then 2 N will have more dividers than N (2 N is a divider and are all the dividers of N ) and thus, certain numbers larger than N (but not larger than 2 N ) must thus be highly made up.

Approximately, so that a number is a highly made up number it must have factors first as small as possible, without being too much of time the same ones.

As follows let us write the decomposition of a number N in factors first:

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

with $p_1 < p_2 < \ cdots < p_k$ first, and nonnull whole exhibitors $c_i$ . Then, the number of dividers of N is exactly:

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

Consequently, so that N is highly made up:

it is necessary that the prime numbers cities are the K more small numbers first (2, 3,5,…); if not, one could replace one of the $p_i$ by the smaller prime number, and obtain a number lower than N having the same number of dividers (for example 10=2×5 can be replaced by 6=2×3, each one has 4 dividers);
it is necessary that $c_1 \ geq c_2 \ geq \ cdots \ geq c_k$ ; if not, by exchanging the two faulty exhibitors one decreases N while preserving the same number of dividers (for example 18=2¹×3² can be replaced by 12=2²×3¹, each one has 6 dividers).

One can as show as it is necessary that C _k = 1, except in two particular cases n=4 and n=36.

The numbers highly made up superiors with 6 are also abundant numbers. Only one glance to the three or four higher dividers of a number highly made up private individual is necessary to confirm this fact. The highly made up numbers are also decomposable in products of Primorielle S.

Many of these numbers is used in the traditional systems measurement, and tends to be used in engineering, because of their use in calculations of complicated Fraction S.

If Q ( X ) represents the quantity of highly made up numbers which are lower or equal to X , then there exist two constants B and C , all two larger than 1, we have

{\ ln X} ^b \ Q (X) \ the {\ ln X} ^c \, \!

The first part of the inequality was proven by Paul Erdős in 1944 and the second part by J. - L. Nicholas in 1988.

Example

See too

senary Binary system
System
duodecimal System
sexagesimal System
Number made up
Table of the dividers

External bonds

Composite Highly Number , MathWorld
Utility
Algorithm
List of the 1000 first

Random links:

Highly made up Number

Example

See too

Related articles

External bonds