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
open all close allBasic Examples (2)
Scope (12)
Numerical Evaluation (5)
DivisorSum works over formal expressions:
DivisorSum[20, f]Exact values are generated at positive integers:
DivisorSum[20, #&]Conditions on divisors can be specified:
DivisorSum[20, #&, # < 5&]DivisorSum[20, f, PrimeQ]DivisorSum[10 ^ 50 + 1, #&]DivisorSum threads elementwise over lists:
DivisorSum[{2, 5, 10}, # ^ 2&]Symbolic Manipulation (7)
TraditionalForm formatting:
DivisorSum[x, f]//TraditionalFormDivisorSum automatically simplifies for polynomial functions:
DivisorSum[n, # ^ 2 + 2# + 5&]Reduce[DivisorSum[n, #&] == 2n - 4 && 0 < n < 40, n, Integers]Solve[DivisorSum[n, EulerPhi] < DivisorSum[n - 1, #&] && 1 < n < 10, n, Integers]FullSimplify[DivisorSum[2 ^ (p - 1)(2 ^ p - 1), #&], Element[2 ^ p - 1, Primes]]DivisorSum[x, MangoldtLambda]DivisorSum[n, MoebiusMu]DivisorSum[n, MoebiusMu[n / #] Log[#] &]GeneratingFunction[DivisorSum[n, Function[m, EulerPhi[m]]], n, x]Applications (8)
Basic Applications (3)
Plot the sum of divisors for the first 100 numbers:
DiscretePlot[DivisorSum[n, #&], {n, 100}]4DivisorSum[n, (-1)^(# - 1) / 2&, OddQ]4DivisorSum[n, Sin[(π #/2)]&]8DivisorSum[n, #&, Mod[#, 4] ≠ 0&]16DivisorSum[n, (-1)^n + ##^3&]DivisorSum[n, 1&]DivisorSum[n, # ^ k&]Number Theory (5)
Compute the Lambert series for Euler's totient function:
GeneratingFunction[DivisorSum[n, Function[m, EulerPhi[m]]], n, x]Compute Jordan's totient function: [more info]
jordanTotient[n_, k_] := DivisorSum[n, # ^ k MoebiusMu[n / #]&];When 
, this is equivalent to Euler's totient function:
AllTrue[Range[100], jordanTotient[#, 1] == EulerPhi[#]&]jordanTotient[#, 2]& /@ Range[20]Compute the twisted divisor sum:
twistedDivisorSum[s_, k_, j_, n_] := DivisorSum[n, DirichletCharacter[k, j, #] * # ^ s&];Table[twistedDivisorSum[s, 6, 2, m], {m, 1, 10}]Define the unitary convolution:
unitaryConvolution[f_, g_, n_, m_] := DivisorSum[m, Evaluate[(f /. n -> #)(g /. n -> (m / #))]&, CoprimeQ[#, m / #]&];Table[unitaryConvolution[n, 1 / n, n, m], {m, 1, 10}]Compute the number of polynomials over 
that are irreducible of degree n:
polynomialCount[p_, n_] := DivisorSum[n, MoebiusMu[n / #]p ^ #&] / n;Irreducible polynomials modulo 5:
Table[polynomialCount[5, n], {n, 1, 20}]Distribution of irreducible polynomials modulo 5:
DiscretePlot[polynomialCount[5, n] / 5 ^ n, {n, 20}, PlotRange -> All]Logarithmic plot of the count for 
:
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:
Total[Divisors[1000] ^ 2]DivisorSum[1000, # ^ 2&]DivisorSigma gives the sum of powers of divisors of an integer:
DivisorSigma[4, 15]DivisorSum[15, # ^ 4&]DivisorSum[n,form] is equivalent to Sum[form[d],{d,Divisors[n]}] for positive n:
n = 28;
form = # ^ 2 + 2#&;DivisorSum[n, form]Sum[form[d], {d, Divisors[n]}]The sum of the prime divisors of a prime number returns the original number:
PrimeQ[11]DivisorSum[11, #&, PrimeQ]Possible Issues (2)
The arguments to DivisorSum are not affected by N:
N[DivisorSum[n, 1 / (# + 1)&]]After evaluation, results may be affected by N:
N[DivisorSum[9, 1 / (# + 1)&]]Only divisors that explicitly yield True on the conditions are used:
DivisorSum[10, f, g]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