For questions on the divisor sum function and its generalizations.

In mathematics, and specifically in number theory, a divisor function is an arithmetic function that returns the number of distinct positive integer divisors of a positive integer.

**Definition:** The divisor function is the sum of positive integers dividing $n$, i.e., $$\sigma(n)=\sum\limits_{d\mid n} d~.$$
As usual, the notation "$d \mid n$ " as the range for a sum or product means that $~d~$ ranges over the positive divisors of $~n~$.

Often a related, more general function $$\sigma_a(n)=\sum\limits_{d\mid n} d^a$$ is studied. Both these functions are multiplicative.

The number of divisors function is given by $$\tau(n)=\sum\limits_{d\mid n} ~1$$

For example, the positive divisors of $~15~$ are $~1,~ 3,~ 5,~$ and $~15~$. So $$\sigma (15)=1+3+5+15=24\qquad \text{and}\qquad \tau(15)=4~.$$

- The divisor function can also be generalized to Gaussian integers.

**References:**

https://en.wikipedia.org/wiki/Divisor_function http://mathworld.wolfram.com/DivisorFunction.html http://sites.millersville.edu/bikenaga/number-theory/divisor-functions/divisor-functions.html