Abundance

How far the divisors of n overshoot twice n: $\sigma(n) - 2n$.

WikipediaAbundant number↗MathWorldAbundance↗OEISA033880↗

Abundance(n)$\sigma(n) - 2n$, where $\sigma$ is the sum of the divisors of $|n|$.

SageMathsigma(abs(n), 1) - 2*abs(n)SymPyabundance(abs(n))

Typed signatures

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

Domain: Number theory

Details

• Positive for an abundant number, zero for a perfect one, negative for a deficient one.
• Defined on nonzero integers, with the sign of $n$ a unit, as in PrimeOmega: $\mathrm{Abundance}(-n)=\mathrm{Abundance}(n)$. At 0 it stays unevaluated.
• $\mathrm{Abundance}(n) = \mathrm{AliquotSum}(n) - n$, which makes IsPerfect the test for 0.
• Neither Wolfram nor Sage names it; SymPy's abundance answers positive integers only.

Plot

Examples

Basic #

Abundant: its proper divisors sum to 16, which is 4 more than 12

live ↗

In$\mathrm{Abundance}(12)$Out$4$

Perfect: $1+2+4+7+14=28$

live ↗

In$\mathrm{Abundance}(28)$Out$0$

Deficient: $\sigma(15)=24$, which is 6 short of 30

live ↗

In$\mathrm{Abundance}(15)$Out$-6$

$\sigma(1)-2=-1$

live ↗

In$\mathrm{Abundance}(1)$Out$-1$

Scope #

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

live ↗

In$\mathrm{Abundance}(-12)$Out$4$

Listable over a list

live ↗

In$\mathrm{Abundance}(\bigl\lbrack6, 12, 15\bigr\rbrack)$Out$\bigl\lbrack0, 4, -6\bigr\rbrack$

Large integers: a power of 2 is always 1 short, $2^{41}-1-2^{41}$

live ↗

In$\mathrm{Abundance}(2^{40})$Out$-1$

Applications #

The smallest odd abundant number: $3^3\cdot5\cdot7$

live ↗

In$\mathrm{Abundance}(945)$Out$30$

The sign classifies: 1 abundant, 0 perfect, -1 deficient

live ↗

In$\mathrm{sgn}(\mathrm{Abundance}(12))$Out$1$

Properties #

$\mathrm{Abundance}(n)=\mathrm{AliquotSum}(n)-n$. See AliquotSum

live ↗

In$\mathrm{Abundance}(12)=\mathrm{AliquotSum}(12)-12$Out$\mathrm{True}$

Perfect numbers have abundance 0, as with 496. See IsPerfect

live ↗

In$\mathrm{Abundance}(496)=0$Out$\mathrm{True}$

Possible issues #

Zero is outside the domain, so it stays unevaluated

live ↗

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

About

Examples	12
Checked against	Wolfram Language 11 agree1 differSageMath 11 agree1 errorSymPy 11 agree1 unevaluatedJulia 11 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

See also: AliquotSum, DivisorSigma, IsPerfect, PerfectNumber