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 allclose all

Basic Examples  (2)

Find the sum of the divisors of :

Plot the sum of divisors for the first 50 numbers:

Scope  (12)

Numerical Manipulation  (5)

DivisorSum works over formal expressions:

Exact values are generated at positive integers:

Conditions on divisors can be specified:

Compute for large numbers:

DivisorSum threads elementwise over lists:

Symbolic Manipulation  (7)

TraditionalForm formatting:

DivisorSum automatically simplifies for polynomial functions:

Reduce expressions:

Solve equations:

Simplify expressions:

Compute sums symbolically:

Generating function:

Applications  (8)

Basic Applications  (3)

Plot the sum of divisors for the first 100 numbers:

Classical identities:

Sum of powers:

Number Theory  (5)

Compute the Lambert series for Euler's totient function:

Compute Jordan's totient function: [more info]

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

Compute the twisted divisor sum:

Define the unitary convolution:

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

Irreducible polynomials modulo 5:

Distribution of irreducible polynomials modulo 5:

Logarithmic plot of the count for :

Properties & Relations  (4)

Use Divisors to compute DivisorSum:

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

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

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

Possible Issues  (2)

The arguments to DivisorSum are not affected by N:

After evaluation, results may be affected by N:

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

Introduced in 2008
 (7.0)