DivisorSigma

The sum of the kth powers of the divisors of n.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDivisor function↗MathWorldDivisorFunction↗WikidataQ915474↗DLMF27.2.E10↗27.2.E9↗27.2.i.p4↗FungrimDivisorSigma↗Fungrim entry7c00e6↗Wolfram LanguageDivisorSigma!↗

DivisorSigma(k, n)sum of the $k$th powers of the divisors of $n$.

SageMathsigma(n, k)SymPydivisor_sigma(n, k)Juliadivisor_sigma(ZZ(n), Int(k))

Typed signatures

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

Domain: Number theory

Details

• $k=0$ gives the divisor count $d(n)$; $k=1$ gives the ordinary sum of divisors.
• Multiplicative: $\sigma_k(mn)=\sigma_k(m)\sigma_k(n)$ whenever $\gcd(m,n)=1$.
• A perfect number $n$ satisfies $\sigma_1(n)=2n$, as with 6 and 28.
• A negative order sums reciprocal powers: $\sigma_{-k}(n) = \sigma_k(n)/n^k$.

Plot

Examples

Basic #

k = 0 gives the number of divisors

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In$\mathrm{DivisorSigma}(0, 20)$Out$6$

k = 1 gives the sum of divisors

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In$\mathrm{DivisorSigma}(1, 20)$Out$42$

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In$\mathrm{DivisorSigma}(2, 20)$Out$546$

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In$\mathrm{DivisorSigma}(2, 6)$Out$50$

Scope #

A negative order sums reciprocal powers: $\sigma_{-1}(6) = 1 + \frac12 + \frac13 + \frac16 = 2$

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In$\mathrm{DivisorSigma}(-1, 6)$Out$2$

Large integers

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In$\mathrm{DivisorSigma}(2, 10^{30})$Out$1\,388\,888\,888\,888\,888\,888\,587\,721\,618\,754\,026\,546\,109\,040\,824\,596\,043\,713\,112\,451$

Negative $k$: $1+\tfrac14+\tfrac1{25}+\tfrac1{100}$

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In$\mathrm{DivisorSigma}(-2, 10)$Out$\frac{13}{10}$

Rational $k$: the square roots of the divisors of 12, summed

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In$\mathrm{DivisorSigma}(\frac{1}{2}, 12)$Out$3+\sqrt{2}+\sqrt{6}+3\sqrt{3}$

Gaussian integers: $1+(1+i)+(1+2i)+(3+i)$, over Divisors

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In$\mathrm{DivisorSigma}(1, 3+i)$Out$6+4i$

GaussianIntegers -> True: squares of $1,1+i,2,3,3+3i,6$

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In$\mathrm{DivisorSigma}(2, 6, \mathrm{Rule}(\mathrm{Over}, \mathrm{GaussianIntegers}))$Out$50+20i$

Listable in $n$

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In$\mathrm{DivisorSigma}(2, \bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)$Out$\bigl\lbrack1, 5, 10, 21, 26\bigr\rbrack$

Symbolic $k$ expands over the divisors

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In$\mathrm{DivisorSigma}(k, 30)$Out$2^{k}+3^{k}+5^{k}+6^{k}+10^{k}+15^{k}+30^{k}+1$

A negative $n$ takes its sign off: $\sigma_1(-12)=\sigma_1(12)=28$, as in Wolfram

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In$\mathrm{DivisorSigma}(1, -12)$Out$28$

Likewise for $k=0$: the 6 divisors of $|n|$

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In$\mathrm{DivisorSigma}(0, -12)$Out$6$

Applications #

6 is a perfect number: $\sigma_1(6)=2\times6$

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In$\mathrm{DivisorSigma}(1, 6)$Out$12$

28 is the next perfect number: $\sigma_1(28)=2\times28$

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In$\mathrm{DivisorSigma}(1, 28)$Out$56$

Perfect numbers have $\sigma(n)=2n$

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In$\mathrm{Filter}(1..30, \operatorname{\_1}↦\mathrm{DivisorSigma}(1, \operatorname{\_1})=2\operatorname{\_1})$Out$\bigl\lbrack6, 28\bigr\rbrack$

Deficient numbers, where $\sigma(n)<2n$

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In$\mathrm{Filter}(1..30, \operatorname{\_1}↦\mathrm{DivisorSigma}(1, \operatorname{\_1})\lt2\operatorname{\_1})$Out$\bigl\lbrack1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29\bigr\rbrack$

Abundant numbers, where $\sigma(n)>2n$

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In$\mathrm{Filter}(1..30, \operatorname{\_1}↦2\operatorname{\_1}\lt\mathrm{DivisorSigma}(1, \operatorname{\_1}))$Out$\bigl\lbrack12, 18, 20, 24, 30\bigr\rbrack$

Iterating the aliquot map from 276, a long-running sequence

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In$\mathrm{NestList}(\operatorname{\_1}↦\mathrm{DivisorSigma}(1, \operatorname{\_1})-\operatorname{\_1}, 276, 20)$Out$\bigl\lbrack276, 396, 696, 1\,104, 1\,872, 3\,770, 3\,790, 3\,050, 2\,716, 2\,772, 5\,964, 10\,164, 19\,628, 19\,684, 22\,876, 26\,404, 30\,044, 33\,796, 38\,780, 54\,628, 54\,684\bigr\rbrack$

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In$(\operatorname{\_1}↦\mathrm{DivisorSigma}(0, \operatorname{\_1}))(\bigl\lbrack2, 3, 5, 6, 7, 8, 10\bigr\rbrack)$Out$\bigl\lbrack2, 2, 2, 4, 2, 4, 4\bigr\rbrack$

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In$(\operatorname{\_1}↦\mathrm{DivisorSigma}(0, \operatorname{\_1}))(\bigl\lbrack1, 4, 9\bigr\rbrack)$Out$\bigl\lbrack1, 3, 3\bigr\rbrack$

Properties #

Multiplicative: $\sigma_1(mn)=\sigma_1(m)\sigma_1(n)$ when $\gcd(m,n)=1$, here for $9$ and $8$

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In$\mathrm{DivisorSigma}(1, 9)\mathrm{DivisorSigma}(1, 8)=\mathrm{DivisorSigma}(1, 72)$Out$\mathrm{True}$

Sum of fifth powers of the divisors of 100

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In$\mathrm{DivisorSigma}(5, 100)$Out$10\,325\,569\,807$

Multiplicative: coprime arguments multiply

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In$\mathrm{DivisorSigma}(3, 18)\mathrm{DivisorSigma}(3, 25)=\mathrm{DivisorSigma}(3, 450)$Out$\mathrm{True}$

$\sigma_{-1}(n)=\sigma(n)/n$, which is 2 for a perfect number

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In$\mathrm{DivisorSigma}(-1, 28)$Out$2$

$\sigma_0$ counts divisors; 1 has one

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In$\mathrm{DivisorSigma}(0, 1)$Out$1$

Known value · Wolfram MathematicalFunctionData, DivisorSigma

A prime has two divisors

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In$\mathrm{DivisorSigma}(0, 2)$Out$2$

Known value · Wolfram MathematicalFunctionData, DivisorSigma

Another prime, two divisors

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In$\mathrm{DivisorSigma}(0, 3)$Out$2$

Known value · Wolfram MathematicalFunctionData, DivisorSigma

Every prime power is deficient

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In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{DivisorSigma}(1, \operatorname{\_1})\lt2\operatorname{\_1}, \mathrm{Filter}(1..100, \mathrm{IsPrimePower}))$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack$

Possible issues #

The trivial base case

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In$\mathrm{DivisorSigma}(1, 1)$Out$1$

About

Examples	31
Checked against	Wolfram Language 29 agree2 differSageMath 17 agree6 errorSymPy 15 agree8 errormathlib4 14 agree8 errorJulia 15 agree7 error
Crosswalk	WikipediaMathWorldWikidataDLMF ×3FungrimFungrim entryWolfram LanguageSageMathSymPyJulia

Implementation

mappedsympypackages/oracle/src/mappings.ts

SymPy's divisor_sigma takes (n, k), the reverse of our (k, n).

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

ArithmeticFunction.sigma k n = the sum of the kth powers of the divisors of n, with sigma k 0 = 0; natural k and n only.

See also: Divisors