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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 $$\sigma (15)=1+3+5+15=24\qquad \text{and}\qquad \tau(15)=4~.$$

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