DivisorSum

The sum of $f(d)$ over the positive divisors $d$ of $n$, optionally only those where $\mathrm{cond}(d)$ holds.

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DivisorSum(n, f)$\sum_{d\mid n} f(d)$

DivisorSum(n, f, cond)$\sum_{d\mid n,\ \mathrm{cond}(d)} f(d)$

Typed signatures

Type	Package	Applies
(integer, function, function?) -> number	number-theory	always

Domain: Number theory

Details

• $f$ and $\mathrm{cond}$ are Function literals over the divisor.
• DivisorSigma, MoebiusMu's divisor-sum identity and Totient's are all instances: see DivisorSigma, MoebiusMu, Totient.

Examples

Basic #

$\sigma_2(20)$. See DivisorSigma

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In$\mathrm{DivisorSum}(20, d↦d^2)$Out$546$

Only the odd divisors: $1+3+5+15$

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In$\mathrm{DivisorSum}(30, d↦d, d↦\mathrm{IsOdd}(d))$Out$24$

Summing the divisors themselves gives $\sigma(30)=72$

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In$\mathrm{DivisorSum}(30, ()↦\operatorname{\_1})$Out$72$

Scope #

With a symbolic function, the sum over divisors is written out

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In$\mathrm{DivisorSum}(20, f)$Out$f(1)+f(2)+f(4)+f(5)+f(10)+f(20)$

A condition restricts the sum to prime divisors

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In$\mathrm{DivisorSum}(20, f, \mathrm{IsPrime})$Out$f(2)+f(5)$

Sum of the divisors of 20

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In$\mathrm{DivisorSum}(20, ()↦\operatorname{\_1})$Out$42$

A condition keeps divisors below 5: $1+2+4$

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In$\mathrm{DivisorSum}(20, ()↦\operatorname{\_1}, ()↦\operatorname{\_1}\lt5)$Out$7$

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In$\mathrm{DivisorSum}(1+10^{50}, ()↦\operatorname{\_1})$Out$\mathrm{DivisorSum}(100\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,001, ()↦\operatorname{\_1})$

Properties #

$\sum_{d\mid n}\mu(d)=0$ for $n>1$

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In$\mathrm{DivisorSum}(12, d↦\mathrm{MoebiusMu}(d))$Out$0$

$\sum_{d\mid n}\varphi(d)=n$

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In$\mathrm{DivisorSum}(36, d↦\mathrm{Totient}(d))$Out$36$

Sum of squared divisors of 1000

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In$\mathrm{DivisorSum}(1\,000, ()↦\operatorname{\_1}^2)$Out$1\,383\,460$

Sum of fourth powers of divisors of 15

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In$\mathrm{DivisorSum}(15, ()↦\operatorname{\_1}^4)$Out$51\,332$

A prime has itself as its only prime divisor

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In$\mathrm{DivisorSum}(11, ()↦\operatorname{\_1}, \mathrm{IsPrime})$Out$11$

Possible issues #

1 has one divisor

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In$\mathrm{DivisorSum}(1, d↦d)$Out$1$

Stays unevaluated for symbolic $n$, even under N

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In$N(\mathrm{DivisorSum}(n, ()↦\frac{1}{\operatorname{\_1}+1}))$Out$\mathrm{DivisorSum}(n, ()↦\frac{1}{\operatorname{\_1}+1})$

With a concrete argument the sum evaluates: $\tfrac12+\tfrac14+\tfrac1{10}$

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In$N(\mathrm{DivisorSum}(9, ()↦\frac{1}{\operatorname{\_1}+1}))$Out$0.85$

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In$\mathrm{DivisorSum}(10, f, g)$Out$\mathrm{DivisorSum}(10, f, {{g}})$

About

Examples	17
Checked against	Wolfram Language 16 agree1 differ
Crosswalk	FungrimWolfram Language

See also: Divisors, DivisorSigma, MoebiusMu, Totient