Divisors

Divisors[n]

gives a list of the integers that divide n.

Details and Options

Examples

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Basic Examples  (1)

The divisors of 1729:

Scope  (2)

For integer input, integer divisors are returned:

For Gaussian integer input, Gaussian divisors are produced:

Divisors threads element‐wise over list arguments:

Options  (3)

GaussianIntegers  (3)

This will produce Gaussian divisors for integer input:

Some primes are also Gaussian primes:

The ratio of Gaussian divisors to integer divisors:

Applications  (3)

Find all perfect numbers less than 10000:

Representation of 25 as sum of two squares:

PowersRepresentations generates an ordered representation:

Number of representations of a number as a sum of four squares:

Computation by SquaresR:

Properties & Relations  (4)

This counts the number of divisors:

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

Similarly, EulerPhi[n]==n∏p|n(1-1/p) where p is prime:

Alternatively, EulerPhi[n]==n∑k|nMoebiusMu[k]/k:

Possible Issues  (1)

Divisors gives all divisors except for multiplication by units; that is, they lie in the first quadrant:

Get all divisors:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_divisors, author="Wolfram Research", title="{Divisors}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Divisors.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_divisors, organization={Wolfram Research}, title={Divisors}, year={1988}, url={https://reference.wolfram.com/language/ref/Divisors.html}, note=[Accessed: 22-November-2024 ]}