AliquotSum

The sum of the proper divisors of n: $\sigma(n) - n$.

WikipediaAliquot sum↗MathWorldRestrictedDivisorFunction↗OEISA001065↗

AliquotSum(n)$s(n)=\sigma(n) - n$, the sum of the divisors of $|n|$ other than $|n|$ itself.

SageMathsigma(abs(n), 1) - abs(n)SymPydivisor_sigma(abs(n)) - abs(n)

Typed signatures

Type	Package	Applies
(integer) -> integer	number-theory	always

Domain: Number theory

Details

• A prime has aliquot sum 1, and $s(1)=0$ since 1 has no proper divisor.
• Defined on nonzero integers, with the sign of $n$ a unit, as in PrimeOmega: $s(-n)=s(n)$. At 0 it stays unevaluated.
• Iterating $s$ gives the aliquot sequence; $n$ is perfect when $s(n)=n$, and $m, n$ are amicable when $s(m)=n$ and $s(n)=m$.
• Neither Wolfram, Sage nor SymPy names it.

Plot

Examples

Basic #

$1+2+3+4+6=16$

live ↗

In$\mathrm{AliquotSum}(12)$Out$16$

Perfect: its proper divisors sum back to 28

live ↗

In$\mathrm{AliquotSum}(28)$Out$28$

A prime has only 1 as a proper divisor

live ↗

In$\mathrm{AliquotSum}(13)$Out$1$

1 has no proper divisor

live ↗

In$\mathrm{AliquotSum}(1)$Out$0$

Scope #

Negative $n$: the sign is a unit, as in PrimeOmega

live ↗

In$\mathrm{AliquotSum}(-12)$Out$16$

Listable over a list

live ↗

In$\mathrm{AliquotSum}(\bigl\lbrack6, 12, 13\bigr\rbrack)$Out$\bigl\lbrack6, 16, 1\bigr\rbrack$

Large integers: $2^{40}-1$, the proper divisors of $2^{40}$

live ↗

In$\mathrm{AliquotSum}(2^{40})$Out$1\,099\,511\,627\,775$

Applications #

Amicable pair: $s(220)=284$

live ↗

In$\mathrm{AliquotSum}(220)$Out$284$

and $s(284)=220$, so $s(s(220))=220$

live ↗

In$\mathrm{AliquotSum}(\mathrm{AliquotSum}(220))$Out$220$

Properties #

$s(n)=\sigma_1(n)-n$. See DivisorSigma

live ↗

In$\mathrm{AliquotSum}(360)=\mathrm{DivisorSigma}(1, 360)-360$Out$\mathrm{True}$

Possible issues #

Zero is outside the domain, so it stays unevaluated

live ↗

In$\mathrm{AliquotSum}(0)$Out$\mathrm{AliquotSum}(0)$

About

Examples	11
Checked against	Wolfram Language 10 agree1 differSageMath 10 agree1 errorSymPy 10 agree1 errormathlib4 7 agree1 unevaluated3 errorJulia 10 agree1 error
Crosswalk	WikipediaMathWorldOEISSageMathSymPy

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no head for it; this is its definition, with the sign of n a unit.

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

Sage has no head for it; this is its definition, with the sign of n a unit.

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

The sum over Nat.properDivisors, so aliquot 0 = 0; natural n only, where ours takes the sign of n as a unit.

See also: Abundance, DivisorSigma, Divisors, IsPerfect