Motivation

Many interesting computational problems, such as those on Project Euler require that one find the sum of proper divisors of a given integer. I had a fairly crude brute-force method for doing this, and was subsequently emailed a comment by Bjarki Ágúst Guðmundsson who runs the site www.mathblog.dk. He pointed me in the direction of this page and provided some sample code illustrating how such an approach runs asymptotically faster than the approach I had been taking. Awesome! I’m going to expand on that a bit here, providing some mathematical proofs behind the claims and providing code for those who may want to take advantage of this.

The Mathematics Behind It All

Let the function $\sigma(n)$ be the sum of divisors for a positive integer $n$. For example,
\[\sigma(6)=1+2+3+6=12.\]

It should seem obvious that for any prime $p$ we have $\sigma(p)=1+p$. What about powers of primes? Let $\alpha\in\mathbb{Z}_+$, and then
\[\sigma(p^\alpha)=1+p+p^2+\cdots+p^\alpha.\]

We’d like to write that in a closed form, i.e., without the “$+\cdots+$”. We use a standard series trick to do that.

\[\begin{align} \sigma(p^\alpha) &= 1+p+p^2+\cdots+p^\alpha\cr p\sigma(p^\alpha) &= p+p^2+p^3+\cdots+p^{\alpha+1}\cr p\sigma(p^\alpha)-\sigma(p^\alpha) &= (p+p^2+\cdots+p^{\alpha+1})-(1+p+\cdots+p^\alpha)\cr p\sigma(p^\alpha)-\sigma(p^\alpha) &= p^{\alpha+1}-1\cr (p-1)\sigma(p^\alpha) &= p^{\alpha+1}-1\cr \sigma(p^\alpha) &=\frac{p^{\alpha+1}-1}{p-1}.\end{align}\]

That solves the problem of finding the sum of divisors for powers of primes. It would be nice if we could show that $\sigma$ is multiplicative on powers of primes, i.e., that $\sigma(p_1^{\alpha_1}p_2^{\alpha_2})=\sigma(p_1^{\alpha_1})\sigma(p_2^{\alpha_2})$. We’ll prove that this is the case, and solve the problem in general along the way.

Proposition: The function $\sigma$ is multiplicative on powers of primes.

Proof: Let $n$ be a positive integer written (uniquely, by the fundamental theorem of arithmetic) as
\[n=\prod_{i=1}^m p_i^{\alpha_i}\]
for $m$ distinct primes $p_i$ with $\alpha_i\in\mathbb{Z}_+$. Any divisor $k$ of $n$ then has the form
\[k=\prod_{i=1}^m p_i^{\beta_i}\]
where each $\beta_i$ satisfies $0\le\beta_i\le\alpha_i$. Then
\[\sigma(n)=\sigma\left(\prod_{i=1}^m p_i^{\alpha_i}\right)\]
is the sum of all divisors $k$ of $n$ and can be written by summing over all possible combinations of the exponents $\beta_i$. There are $\prod_{i=1}^m \alpha_i$ combinations, and we can form their sum and simplify it as follows.
\[\begin{align}\sigma(n) &= \sum_{1\le i\le m,\ 0\le\beta_i\le\alpha_i}p_1^{\beta_i}p_2^{\beta_2}\cdots p_m^{\beta_m}\cr &= \sum_{\beta_1=0}^{\alpha_1}p_1^{\beta_1}\left(\sum_{2\le i\le m,\ 0\le\beta_i\le\alpha_i}p_2^{\beta_2}p_3^{\beta_3}\cdots p_m^{\beta_m}\right)\cr &= \sum_{\beta_1=0}^{\alpha_1}p_1^{\beta_1}\sum_{\beta_2=0}^{\alpha_2}p_2^{\beta_2}\left(\sum_{3\le i\le m,\ 0\le\beta_i\le\alpha_i}p_3^{\beta_3}p_4^{\beta_4}\cdots p_m^{\beta_m}\right) \cr &= \vdots\cr &=\sum_{\beta_1=0}^{\alpha_1}p_1^{\beta_1}\sum_{\beta_2=0}^{\alpha_2}p_2^{\beta_2}\sum_{\beta_3=0}^{\alpha_3}p_3^{\beta_3}\ \cdots\ \sum_{\beta_m=0}^{\alpha_m}p_m^{\beta_m}\cr &=\sigma(p_1^{\alpha_1})\sigma(p_2^{\alpha_2})\cdots\sigma(p_m^{\alpha_m}).\end{align}\]
This completes the proof. Q.E.D.

Thus, we now have a formula for the sum of divisors of an arbitrary positive integer $n$ using the factorization of $n$. Namely,
\[\sigma(n)=\sigma\left(\prod_{i=1}^m p_i^{\alpha_i}\right)=\prod_{i=1}^m\left(\frac{p_i^{\alpha_i+1}-1}{p_i-1}\right).\]


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