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DivisorSum[n,form]

represents the sum of form[i] for all i that divide n.

DivisorSum[n,form,cond]

includes only those divisors for which cond[i] gives True.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Symbolic Manipulation  
Applications  
Basic Applications  
Number Theory  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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DivisorSum

DivisorSum[n,form]

represents the sum of form[i] for all i that divide n.

DivisorSum[n,form,cond]

includes only those divisors for which cond[i] gives True.

Details

  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • n can be symbolic or a positive integer.
  • form and cond must be Function objects.
  • DivisorSum[n,form] is equivalent to Sum[form[d],{d,Divisors[n]}] for positive n.
  • DivisorSum[n,form,cond] is automatically simplified when n is a positive integer.
  • DivisorSum[n,form] is automatically simplified when form is a polynomial function.

Examples

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Basic Examples  (2)

Find the sum of the divisors of :

Wolfram Language code: DivisorSum[30, #&]

Plot the sum of divisors for the first 50 numbers:

Wolfram Language code: DiscretePlot[DivisorSum[n, #&], {n, 50}]

Scope  (12)

Numerical Evaluation  (5)

DivisorSum works over formal expressions:

Wolfram Language code: DivisorSum[20, f]

Exact values are generated at positive integers:

Wolfram Language code: DivisorSum[20, #&]

Conditions on divisors can be specified:

Wolfram Language code: DivisorSum[20, #&, # < 5&]
Wolfram Language code: DivisorSum[20, f, PrimeQ]

Compute for large numbers:

Wolfram Language code: DivisorSum[10 ^ 50 + 1, #&]

DivisorSum threads elementwise over lists:

Wolfram Language code: DivisorSum[{2, 5, 10}, # ^ 2&]

Symbolic Manipulation  (7)

TraditionalForm formatting:

Wolfram Language code: DivisorSum[x, f]//TraditionalForm

DivisorSum automatically simplifies for polynomial functions:

Wolfram Language code: DivisorSum[n, # ^ 2 + 2# + 5&]

Reduce expressions:

Wolfram Language code: Reduce[DivisorSum[n, #&] == 2n - 4 && 0 < n < 40, n, Integers]

Solve equations:

Wolfram Language code: Solve[DivisorSum[n, EulerPhi] < DivisorSum[n - 1, #&] && 1 < n < 10, n, Integers]

Simplify expressions:

Wolfram Language code: FullSimplify[DivisorSum[2 ^ (p - 1)(2 ^ p - 1), #&], Element[2 ^ p - 1, Primes]]

Compute sums symbolically:

Wolfram Language code: DivisorSum[x, MangoldtLambda]
Wolfram Language code: DivisorSum[n, MoebiusMu]
Wolfram Language code: DivisorSum[n, MoebiusMu[n / #] Log[#] &]

Generating function:

Wolfram Language code: GeneratingFunction[DivisorSum[n, Function[m, EulerPhi[m]]], n, x]

Applications  (8)

Basic Applications  (3)

Plot the sum of divisors for the first 100 numbers:

Wolfram Language code: DiscretePlot[DivisorSum[n, #&], {n, 100}]

Classical identities:

Wolfram Language code: 4DivisorSum[n, (-1)^(# - 1) / 2&, OddQ]
Wolfram Language code: 4DivisorSum[n, Sin[(π #/2)]&]
Wolfram Language code: 8DivisorSum[n, #&, Mod[#, 4] ≠ 0&]
Wolfram Language code: 16DivisorSum[n, (-1)^n + ##^3&]

Sum of powers:

Wolfram Language code: DivisorSum[n, 1&]
Wolfram Language code: DivisorSum[n, # ^ k&]

Number Theory  (5)

Compute the Lambert series for Euler's totient function:

Wolfram Language code: GeneratingFunction[DivisorSum[n, Function[m, EulerPhi[m]]], n, x]

Compute Jordan's totient function: [more info]

Wolfram Language code: jordanTotient[n_, k_] := DivisorSum[n, # ^ k MoebiusMu[n / #]&];

When , this is equivalent to Euler's totient function:

Wolfram Language code: AllTrue[Range[100], jordanTotient[#, 1] == EulerPhi[#]&]
Wolfram Language code: jordanTotient[#, 2]& /@ Range[20]

Compute the twisted divisor sum:

Wolfram Language code: twistedDivisorSum[s_, k_, j_, n_] := DivisorSum[n, DirichletCharacter[k, j, #] * # ^ s&];
Wolfram Language code: Table[twistedDivisorSum[s, 6, 2, m], {m, 1, 10}]

Define the unitary convolution:

Wolfram Language code: unitaryConvolution[f_, g_, n_, m_] := DivisorSum[m, Evaluate[(f /. n -> #)(g /. n -> (m / #))]&, CoprimeQ[#, m / #]&];
Wolfram Language code: Table[unitaryConvolution[n, 1 / n, n, m], {m, 1, 10}]

Compute the number of polynomials over that are irreducible of degree n:

Wolfram Language code: polynomialCount[p_, n_] := DivisorSum[n, MoebiusMu[n / #]p ^ #&] / n;

Irreducible polynomials modulo 5:

Wolfram Language code: Table[polynomialCount[5, n], {n, 1, 20}]

Distribution of irreducible polynomials modulo 5:

Wolfram Language code: DiscretePlot[polynomialCount[5, n] / 5 ^ n, {n, 20}, PlotRange -> All]

Logarithmic plot of the count for :

Wolfram Language code: DiscretePlot[{polynomialCount[2, n], polynomialCount[3, n], polynomialCount[5, n]}, {n, 20}, ScalingFunctions -> "Log", PlotLegends -> {"p=2", "p=3", "p=5"}]

Properties & Relations  (4)

Use Divisors to compute DivisorSum:

Wolfram Language code: Total[Divisors[1000] ^ 2]
Wolfram Language code: DivisorSum[1000, # ^ 2&]

DivisorSigma gives the sum of powers of divisors of an integer:

Wolfram Language code: DivisorSigma[4, 15]
Wolfram Language code: DivisorSum[15, # ^ 4&]

DivisorSum[n,form] is equivalent to Sum[form[d],{d,Divisors[n]}] for positive n:

Wolfram Language code: n = 28; form = # ^ 2 + 2#&;
Wolfram Language code: DivisorSum[n, form]
Wolfram Language code: Sum[form[d], {d, Divisors[n]}]

The sum of the prime divisors of a prime number returns the original number:

Wolfram Language code: PrimeQ[11]
Wolfram Language code: DivisorSum[11, #&, PrimeQ]

Possible Issues  (2)

The arguments to DivisorSum are not affected by N:

Wolfram Language code: N[DivisorSum[n, 1 / (# + 1)&]]

After evaluation, results may be affected by N:

Wolfram Language code: N[DivisorSum[9, 1 / (# + 1)&]]

Only divisors that explicitly yield True on the conditions are used:

Wolfram Language code: DivisorSum[10, f, g]
Wolfram Research (2008), DivisorSum, Wolfram Language function, https://reference.wolfram.com/language/ref/DivisorSum.html.

Text

Wolfram Research (2008), DivisorSum, Wolfram Language function, https://reference.wolfram.com/language/ref/DivisorSum.html.

CMS

Wolfram Language. 2008. "DivisorSum." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DivisorSum.html.

APA

Wolfram Language. (2008). DivisorSum. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DivisorSum.html

BibTeX

@misc{reference.wolfram_2026_divisorsum, author="Wolfram Research", title="{DivisorSum}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DivisorSum.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_divisorsum, organization={Wolfram Research}, title={DivisorSum}, year={2008}, url={https://reference.wolfram.com/language/ref/DivisorSum.html}, note=[Accessed: 13-June-2026]}