Divisor function
Under the divisor sum σ of a natural number is defined as the sum of all the divisors of that number including the number itself
Example:
In many problems of number theory divider sums play a role, eg in the perfect numbers and amicable numbers.
- 2.1 Set 1: divider sum of a prime number
- 2.2 Set 2: divider sum of the power of a prime
- 2.3 Clause 3: divider sum of the product of two primes
- 2.4 Clause 4: generalization of Theorem 2 and Theorem 3
Definitions
Definition 1: sum of all divisors
Are all factors of the natural number n, then it is called the dividing this sum of n 1 and n divider itself, ie in the set of the divider. The σ function is called the divisor sum function and is a number-theoretic function.
The above example can now be written:
Definition 2: sum of the proper divisors
The sum of the proper divisor of the natural number n is the sum of the factor of n without the number n is even, and we refer to this sum with.
Example:
Obviously, the relation:
Definition 3: deficient, abundant, completely
A natural number n> 1 is called
Examples:
Properties of the divisor sum
Set 1: divider sum of a prime number
Let n be a prime number. Then:
Proof: Since n is a prime number, n is 1 and the single dividers. It follows the assertion.
Set 2: divider sum of the power of a prime
Let n be a prime number. Then:
Proof: Since n is prime, are the divisors of nk: n0, n1, ..., nk. The sum is a geometric series. From the sum formula for a geometric series immediately follows the assertion.
Example:
Set 3: divider sum of the product of two primes
Let a and b are distinct primes. Then:
Proof: The number has four different from splitter 1, A, B and AB. It follows:
Example:
Set 4: generalization of Theorem 2 and Theorem 3
Be different prime numbers and natural numbers. Furthermore, it is. Then:
Set of Thabit
With the help of Theorem 4 one can prove the theorem by Thabit from the field of amicable numbers. The sentence reads:
For a fixed natural number n, let x = 3.2 s -1, y = 3.2 s -1-1, and z = 9.22 n-1 -1.
When x, y and z are prime numbers greater than 2, then the two numbers A = 2 n · x · y · z and b = 2n friends, and that.
Proof:
Analog shows you.
Divider sum as a finite series
For every natural number is the divisor function can be represented as a series without that is made to the divisibility properties of explicit reference:
Proof: The function
Is 1 if a divisor of, otherwise they will remain zero. First applies
The numerator in the last expression is always zero when goes. The denominator can only be zero if a divisor of is. But then
Only in this case, as claimed above.
Multiplying now and summed with the product of all values to a post so arises only to the sum, if a divisor of is. But this is exactly the definition of the general Divisorfunktion
The special case is the sum of simple divider.