DivisorSigma works over integers:

Negative powers:

Rational powers:

Gaussian integers:

Compute for large integers:

DivisorSigma threads elementwise over lists:

TraditionalForm formatting:

Reduce expressions:

Solve equations:

Simplify expressions:

Use DivisorSigma in a sum:

DirichletTransform:

Compute for symbolic arguments:

Generate a function:

Find the sum of the divisors of over integers:

Gaussian integers:

In general, DivisorSigma[d,n]=∑_k|nk^d:

The ratio of Gaussian divisors to integer divisors:

Plot DivisorSigma with log-scaled values:

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:

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 :