Divisors

All positive divisors of n, in increasing order.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDivisor↗MathWorldDivisor↗WikidataQ50708↗Encyclopedia of MathematicsDivisor_(of_an_integer_or_of_a_polynomial)↗Britannicatopic/divisor↗Fungrim entry08ff0b↗Wolfram LanguageDivisors!↗Rosetta CodeProper divisors↗

Divisors(n)all positive divisors of $n$, increasing.

SageMathenumeratio_divisors(n)SymPydivisors(n)Oscarsort(divisors(ZZ(n)))Juliasort(divisors(ZZ(n)))

Divisors(n, Over -> GaussianIntegers)the first-quadrant divisors in $\mathbb{Z}[i]$, by real part then imaginary

Typed signatures

Type	Package	Applies
(integer) -> list<integer>	compute-engine	always
(number | quadratic_integer, any*) -> list	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• Includes both 1 and $n$ itself; a prime's only divisors are those two.
• The count of divisors, $d(n)$, equals $\sigma_0(n)$. See DivisorSigma.
• Highly composite numbers (like 720) pack unusually many divisors relative to their size.
• compute-engine only returns ordinary positive divisors.
• Threads element-wise over a list, as Wolfram's Listable heads do.

Examples

19 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{Divisors}(1\,729)$Out$\bigl\lbrack1, 7, 13, 19, 91, 133, 247, 1\,729\bigr\rbrack$

live ↗

In$\mathrm{Divisors}(20)$Out$\bigl\lbrack1, 2, 4, 5, 10, 20\bigr\rbrack$

A prime's only divisors are 1 and itself

live ↗

In$\mathrm{Divisors}(13)$Out$\bigl\lbrack1, 13\bigr\rbrack$

live ↗

In$\mathrm{Divisors}(6)$Out$\bigl\lbrack1, 2, 3, 6\bigr\rbrack$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

live ↗

In$\mathrm{Divisors}(\bigl\lbrack4, 6\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 2, 4\bigr\rbrack, \bigl\lbrack1, 2, 3, 6\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{Divisors}(3+4i)$Out$\bigl\lbrack1, 2+i, 3+4i\bigr\rbrack$

$13 = (2 + 3i)(3 - 2i)$ splits, so it gains two divisors

live ↗

In$\mathrm{Divisors}(13, \mathrm{Rule}(\mathrm{Over}, \mathrm{GaussianIntegers}))$Out$\bigl\lbrack1, 2+3i, 3+2i, 13\bigr\rbrack$

$6+4i=-i(1+i)^2(3+2i)$

live ↗

In$\mathrm{Divisors}(6+4i)$Out$\bigl\lbrack1, 1+i, 1+5i, 2, 3+2i, 6+4i\bigr\rbrack$

Listable: threads over a list

live ↗

In$\mathrm{Divisors}(\bigl\lbrack605, 871, 824\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 5, 11, 55, 121, 605\bigr\rbrack, \bigl\lbrack1, 13, 67, 871\bigr\rbrack, \bigl\lbrack1, 2, 4, 8, 103, 206, 412, 824\bigr\rbrack\bigr\rbrack$

The smallest odd prime: just 1 and 3

live ↗

In$\mathrm{Divisors}(3)$Out$\bigl\lbrack1, 3\bigr\rbrack$

Applications #

Sum of divisors of 20 not divisible by 4, scaled by 8

live ↗

In$8(\sum\mathrm{Filter}(\mathrm{Divisors}(20), \operatorname{\_1}↦¬(4\mid\operatorname{\_1})))$Out$144$

live ↗

In$\mathrm{Filter}(1..9\,999, \operatorname{\_1}↦\sum\mathrm{Divisors}(\operatorname{\_1})=2\operatorname{\_1})$Out$\mathrm{Filter}(1..9\,999, \operatorname{\_1}↦\sum\mathrm{Divisors}(\operatorname{\_1})=2\operatorname{\_1})$

Properties #

The divisor count $d(n)$ equals $\sigma_0(n)$. See DivisorSigma

live ↗

In$\mathrm{Length}(\mathrm{Divisors}(20))=\mathrm{DivisorSigma}(0, 20)$Out$\mathrm{True}$

Smallest prime: just 1 and 2

live ↗

In$\mathrm{Divisors}(2)$Out$\bigl\lbrack1, 2\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

A prime square has three divisors

live ↗

In$\mathrm{Divisors}(4)$Out$\bigl\lbrack1, 2, 4\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

A prime has only 1 and itself

live ↗

In$\mathrm{Divisors}(5)$Out$\bigl\lbrack1, 5\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

A prime contributes only the trivial divisors

live ↗

In$\mathrm{Divisors}(7)$Out$\bigl\lbrack1, 7\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Prime cube: powers of 2 up to 8

live ↗

In$\mathrm{Divisors}(8)$Out$\bigl\lbrack1, 2, 4, 8\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Prime square: 1, 3, 9

live ↗

In$\mathrm{Divisors}(9)$Out$\bigl\lbrack1, 3, 9\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Product of two primes gives four divisors

live ↗

In$\mathrm{Divisors}(10)$Out$\bigl\lbrack1, 2, 5, 10\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

11 is prime, so two divisors

live ↗

In$\mathrm{Divisors}(11)$Out$\bigl\lbrack1, 11\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Six divisors, listed in increasing order

live ↗

In$\mathrm{Divisors}(12)$Out$\bigl\lbrack1, 2, 3, 4, 6, 12\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

$14 = 2\cdot 7$: each subset of primes gives a divisor

live ↗

In$\mathrm{Divisors}(14)$Out$\bigl\lbrack1, 2, 7, 14\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

An odd semiprime, $3\cdot 5$

live ↗

In$\mathrm{Divisors}(15)$Out$\bigl\lbrack1, 3, 5, 15\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Powers of 2 up to 16

live ↗

In$\mathrm{Divisors}(16)$Out$\bigl\lbrack1, 2, 4, 8, 16\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Two divisors mark 17 as prime

live ↗

In$\mathrm{Divisors}(17)$Out$\bigl\lbrack1, 17\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

$18 = 2\cdot 3^2$ has $2\cdot 3 = 6$ divisors

live ↗

In$\mathrm{Divisors}(18)$Out$\bigl\lbrack1, 2, 3, 6, 9, 18\bigr\rbrack$

Known value · Wolfram MathematicalFunctionData, Divisors

Possible issues #

1 has exactly one divisor: itself

live ↗

In$\mathrm{Divisors}(1)$Out$\bigl\lbrack1\bigr\rbrack$

Only positive divisors, even for negative $n$

live ↗

In$\mathrm{Divisors}(-6)$Out$\bigl\lbrack1, 2, 3, 6\bigr\rbrack$

Neat examples #

720 is highly composite: 30 divisors, more than any smaller number

live ↗

In$\mathrm{Divisors}(720)$Out$\bigl\lbrack1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720\bigr\rbrack$

About

Examples	49
Checked against	Wolfram Language 48 agree1 differSageMath 37 agree1 differ1 errorSymPy 29 agree5 differ5 errormathlib4 22 agree1 differ6 errorOscar 24 agreeJulia 26 agree12 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsBritannicaFungrim entryWolfram LanguageSageMathSymPyOscarJuliaRosetta Code

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy's divisors is over the integers (positive divisors of |n|); a Gaussian integer is not one, and the Over -> GaussianIntegers form has no counterpart.

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Nat.divisors sorted ascending into a list; natural n only, with divisors 0 = the empty list.

See also: FactorInteger, DivisorSigma