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DivisorSigma

DivisorSigma
gives the divisor function .
  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • is the sum of the ^(th) powers of the divisors of n.
Sum of divisors:
Sum of squares of divisors:
Plot the number of divisors:
Sums of divisors:
Sums of squared divisors:
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Click for copyable input
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Sum of divisors:
In[2]:=
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Sum of squares of divisors:
In[3]:=
Click for copyable input
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Plot the number of divisors:
In[1]:=
Click for copyable input
Out[1]=
Sums of divisors:
In[2]:=
Click for copyable input
Out[2]=
Sums of squared divisors:
In[3]:=
Click for copyable input
Out[3]=
DivisorSigma threads element-wise over lists:
TraditionalForm formatting:
The first argument can be symbolic:
Include complex divisors:
By default, only integer divisors are included for integer input:
This also includes complex divisors:
Plot the running average of the number of divisors with its asymptotic value:
Find :
Compute an iterated aliquot sum:
Compare the number of divisors with the totient:
Generate values using the definition:
Compute using DivisorSigma:
Use FullSimplify to simplify expressions containing DivisorSigma:
DivisorSigma is a multiplicative function:
Generating function:
With GaussianIntegers->True, the naive definition does not give the correct result:
To make DivisorSigma a multiplicative function, a definition involving factors is used:
Show the evolution of the limit :
Fourier transform of the divisor function in the complex plane:
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