Built-in Mathematica Symbol

DivisorSigma

DivisorSigma[k, n]
gives the divisor function

.

MORE INFORMATION

Integer mathematical function, suitable for both symbolic and numerical manipulation.

is the sum of the k powers of the divisors of n.

DivisorSigma[k, n, GaussianIntegers->True] includes divisors that are Gaussian integers.

DivisorSigma automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Sum of divisors:

Sum of squares of divisors:

Plot the number of divisors:

Sums of divisors:

Sums of squared divisors:

Sum of divisors:

Sum of squares of divisors:

In[3]:=

Out[3]=

Plot the number of divisors:

Sums of divisors:

Sums of squared divisors:

In[3]:=

Out[3]=

Scope (2)

DivisorSigma threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (2)

The first argument can be symbolic:

Include complex divisors:

Options (1)

By default, only integer divisors are included for integer input:

This also includes complex divisors:

Applications (4)

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

Find perfect numbers:

Compute an iterated aliquot sum:

Compare the number of divisors with the totient:

Properties & Relations (4)

Generate values using the definition:

Compute using DivisorSigma:

Use FullSimplify to simplify expressions containing DivisorSigma:

DivisorSigma is a multiplicative function:

Generating function:

Possible Issues (1)

With GaussianIntegers->True, the naive definition does not give the correct result:

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

Neat Examples (2)

Show the evolution of the limit

:

Fourier transform of the divisor function in the complex plane: