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I was playing around on Desmos with a function that computed the sums of proper divisors of an integer and found an interesting pattern regarding the "slopes" of the graph: Graph of integers to sums of their divisors

As you can see, there appear to be several ratios of integers to sums of their proper divisors that are "shared" by enough integers to be immediately visible as a pattern. I was wondering if there was any known proof, identity, or anything that explains these lines. Thanks!

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    $\begingroup$ Easier to see with the sum of all divisors, not just proper divisors, since the sum of all divisors is a multiplicative function. In particular, if $\sigma(n)$ is the sum of all divisors, if $p$ is prime, $\sigma(p^k)/p^k$ is $$\frac{1-1/p^{k+1}}{1-1/p}=\frac{p^{k+1}-1}{p^{k}(p-1)}.$$ and more generally, the value of $\sigma(n)/n$ will be a product of such terms. $\endgroup$
    – Thomas Andrews
    Commented Jun 3 at 1:13
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    $\begingroup$ Welcome to Math SE. For any relatively large prime $p$, consider relatively small multiples of $p$. For example, let $f(x)$ be the sum of proper divisors of $x$, and $g(x) = \frac{f(x)}{x}$. Then $g(2p)=\frac{1+2+p}{2p}\approx\frac{1}{2}$, $g(3p)=\frac{1+3+p}{3p}\approx\frac{1}{3}$, $g(4p)=\frac{1+2+4+p+2p}{4p}\approx\frac{3}{4}$, etc. I suspect this is a significant factor that causes those integer ratios line patterns that I believe your question is inquiring about. If you're asking about something else instead, please clarify. $\endgroup$
    – John Omielan
    Commented Jun 3 at 1:20

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