Divisor function

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The divisor function is denoted $\sigma_k(n)$ and is defined as the sum of the $k$th powers of the divisors of $n$. Thus $\sigma_k(n) = \sum_{d|n}d^k = d_1 + d_2 + \cdots + d_r$ where the $d_i$ are the divisors of $n$.

Number of divisors

Letting $k=0$ makes all of the terms in $d_1 + d_2 + \cdots + d_r$ equal to 1. Thus, The value of $\sigma_0(n)$ is simply the number of divisors of $n$.

Using combinatorics, we can find how many divisors $n$ has if the prime factorization of $n$ is $p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$. Any divisor of $n$ must be of the form $p_1^{f_1}p_2^{f_2} \cdots p_k^{e_k}$ where the $\displaystyle f_i$ are integers such that $0\le f_i \le e_i$ for $i = 1,2,\ldots, k$. Thus, the number of divisors of $n$ is $(e_1+1)(e_2+1)\cdots (e_k+1)$.

Sum of divisors

The sum of the divisors, or $\sigma_1(n)$, is given by

$\sigma_1(n) = (1 + p_1 + p_1^2 +\cdots p_1^{e_1})(1 + p_2 + p_2^2 + \cdots + p_2^{e_2}) \cdots (1 + p_n + p_n^2 + \cdots + p_n^{e_n}).$

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See also