Highly composite number

A highly composite number ( engl. highly composite number, in short: HCN) is a positive integer that has more dividers than any smaller positive integer. Such figures are a kind of counterpart to the prime because of their maximum divisibility. They were examined with the first from the Indian mathematician Ramanujan S..

The first twenty highly composite numbers:

Properties

Construction

Every positive integer is structured as follows:

The prime. The exponents are nonzero integers. For there is the empty product.

The number of divisors of a positive natural number can be determined using the following formula:

For highly composite numbers applies:

• The primes are exactly the first prime numbers, because every exuberant prime would make it possible to construct a smaller divider with the same number.
• The sequence of exponents is non- increasing, ie, for otherwise it would be possible by interchanging exponent, a smaller construct with the same divisor number.

These two features are necessary, but not sufficient. So must except and be the last exponent.

Example:

Divisibility

The ability to have as many dividers, offers certain practical advantages and is therefore often sought deliberately. Thus, the angle degree system is based on 360 ° on a highly composite number. Also the hours to 24, minutes, and seconds of 60 units, and the old coinage of Charlemagne with the relationship a pound of silver equal to 240 pennies or denarii are mentioned here. Even in Prussia from 1821 to 1873 a thaler equal to 360 pennies.

Ramanujan and highly composite numbers

As one of the first mathematicians of the Indians S. Ramanujan worked extensively with highly composite numbers. He found the above rule of non- increasing exponent. The rule can be used to construct highly composite numbers. Ramanujan himself made ​​a list of over one hundred of the first highly composite numbers. But he failed to see a single one. Today, online lists with over a hundred thousand numbers to find this number sequence.

Curious thing

Although highly composite numbers with their capacity to the maximum divisibility represent a counterpoint to the primes, they are mathematical - like all natural numbers - as fast primes. For example, the highly divisible numbers 24 and 60 are categorized as almost-prime 4 -th order.