Divisor function
The divisor function is denoted and is defined as the sum of the
th powers of the divisors of
. Thus
where the
are the divisors of
.
Number of divisors
Letting makes all of the terms in
equal to 1. Thus, The value of
is simply the number of divisors of
.
Using combinatorics, we can find how many divisors has if the prime factorization of
is
. Any divisor of
must be of the form
where the
are integers such that
for
. Thus, the number of divisors of
is
.
Sum of divisors
The sum of the divisors, or , is given by
![$\sigma_1(n) = (1 + p_1 + p_1^2 +\cdots p_1^{e_1})(1 + p_2 + p_2^2 + \cdots + p_2^{e_2}) \cdots (1 + p_n + p_n^2 + \cdots + p_n^{e_n}).$](http://latex.artofproblemsolving.com/a/0/1/a01fdafba55f143dafb4631b35b41fea26a857b2.png)
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