# DivisorSigma

DivisorSigma[k,n]

gives the divisor function .

# Details and Options • DivisorSigma is also known as the divisor function or sumofdivisors function.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• DivisorSigma[k,n] is the sum of the k powers of the divisors of n.
• For a number with a unit and primes, DivisorSigma[k,n] returns  .
• • With the setting , DivisorSigma includes divisors that are Gaussian integers.
• DivisorSigma[k,m+In] automatically works over Gaussian integers.

# Examples

open allclose all

## Basic Examples(2)

Find the divisors of :

The sum of divisors:

The sum of squares of divisors:

Plot DivisorSigma with log-scaled values:

## Scope(12)

### Numerical Manipulation(4)

DivisorSigma works over integers:

Negative powers:

Rational powers:

Gaussian integers:

Compute for large integers:

DivisorSigma threads elementwise over lists:

### Symbolic Manipulation(8)

TraditionalForm formatting:

Reduce expressions:

Solve equations:

Simplify expressions:

Use DivisorSigma in a sum:

Compute for symbolic arguments:

Generate a function:

## Options(1)

### GaussianIntegers(1)

Find the sum of the divisors of over integers:

Gaussian integers:

## Applications(13)

### Basic Applications(3)

In general, DivisorSigma[d,n]=k|nkd:

The ratio of Gaussian divisors to integer divisors:

Plot DivisorSigma with log-scaled values:

### Special Sequences(4)

Recognize perfect numbers, numbers n such that the sum of their divisors is equal to :

Deficient numbers, numbers n such that the sum of their divisors is smaller than :

Abundant numbers, numbers n such that the sum of their divisors is greater than :

Recognize highly composite numbers: [more info]

Recognize amicable numbers, two different numbers such that the sum of the proper divisors of each is equal to the other number:

Recognize -multiperfect numbers, numbers such that the sum of their divisors is equal to :

The first -perfect number is :

Numbers that are -perfect are called perfect numbers:

### Number Theory(6)

If n is a power of , then the sum of the divisors of n equals , which makes n almost perfect:

The number of the divisors is odd if and only if the number is a perfect square:

Compare the number of divisors with Euler's totient function:

Plot the running average of the number of divisors with its asymptotic value:

Compute an iterated aliquot sum:

Show the evolution of the limit :

## Properties & Relations(6)

DivisorSigma is the sum of the powers of the divisors:

Use DivisorSum to find the sum of divisors:

DivisorSigma is a multiplicative function:

The reciprocals of the divisors of a perfect number n must add up to :

The sum of divisors of a prime power n is less than 2n:

For a prime number p, the number of the divisors is :

The sum of the divisors is :

The number of divisors of is :

Use DivisorSigma to find the product of divisors:

## Possible Issues(1)

With , the naive definition does not give the correct result:

To make DivisorSigma a multiplicative function, a definition involving factors is used:

## Neat Examples(4)

Plot the arguments of the Fourier transform of DivisorSigma:

Plot the absolute values of the Fourier transform of DivisorSigma:

Plot the arguments of the Fourier transform of DivisorSigma:

Plot the Ulam spiral of the mean of the divisors: