DivisorSigma

DivisorSigma[k,n]

gives the divisor function .

Details and Options

  • DivisorSigma is also known as the divisor function or sum‐of‐divisors function.
  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • DivisorSigma[k,n] is the sum of the k ^(th) powers of the divisors of n.
  • For a number with a unit and primes, DivisorSigma[k,n] returns .
  • With the setting GaussianIntegers->True, DivisorSigma includes divisors that are Gaussian integers.
  • DivisorSigma[k,m+In] automatically works over Gaussian integers.

Examples

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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:

DirichletTransform:

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 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  (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:

Wolfram Research (1988), DivisorSigma, Wolfram Language function, https://reference.wolfram.com/language/ref/DivisorSigma.html.

Text

Wolfram Research (1988), DivisorSigma, Wolfram Language function, https://reference.wolfram.com/language/ref/DivisorSigma.html.

CMS

Wolfram Language. 1988. "DivisorSigma." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DivisorSigma.html.

APA

Wolfram Language. (1988). DivisorSigma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DivisorSigma.html

BibTeX

@misc{reference.wolfram_2022_divisorsigma, author="Wolfram Research", title="{DivisorSigma}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/DivisorSigma.html}", note=[Accessed: 31-January-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_divisorsigma, organization={Wolfram Research}, title={DivisorSigma}, year={1988}, url={https://reference.wolfram.com/language/ref/DivisorSigma.html}, note=[Accessed: 31-January-2023 ]}