Divisors

Divisors[n]

gives a list of the integers that divide n.

Details and Options

Examples

open allclose all

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 elementwise 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]==np|n(1-1/p) where p is prime:

Alternatively, EulerPhi[n]==nk|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:

Introduced in 1988
 (1.0)